diff --git a/src/category-theory/adjunctions-large-precategories.lagda.md b/src/category-theory/adjunctions-large-precategories.lagda.md index 0f3ddc0d82..d98a4ff347 100644 --- a/src/category-theory/adjunctions-large-precategories.lagda.md +++ b/src/category-theory/adjunctions-large-precategories.lagda.md @@ -160,7 +160,7 @@ module _ ( map-inv-equiv-is-adjoint-pair-Large-Precategory H X2 Y2) naturality-inv-equiv-is-adjoint-pair-Large-Precategory H {X1 = X1} {X2} {Y1} {Y2} f g = - coherence-square-maps-inv-equiv-horizontal + horizontal-inv-equiv-coherence-square-maps ( equiv-is-adjoint-pair-Large-Precategory H X1 Y1) ( λ h → comp-hom-Large-Precategory D diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md index dbcbe3991e..cd43fb75bf 100644 --- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md @@ -85,7 +85,7 @@ open import univalent-combinatorics.standard-finite-types ## Ideas The delooping of a group homomorphism `f : G → H` is a pointed map -`Bf : BG → BH` equiped with an homotopy witnessing that the following square +`Bf : BG → BH` equiped with a homotopy witnessing that the following square commutes : ```text diff --git a/src/foundation-core.lagda.md b/src/foundation-core.lagda.md index 5244d7271e..7f25f3e438 100644 --- a/src/foundation-core.lagda.md +++ b/src/foundation-core.lagda.md @@ -10,6 +10,7 @@ open import foundation-core.cartesian-product-types public open import foundation-core.coherently-invertible-maps public open import foundation-core.commuting-prisms-of-maps public open import foundation-core.commuting-squares-of-homotopies public +open import foundation-core.commuting-squares-of-identifications public open import foundation-core.commuting-squares-of-maps public open import foundation-core.commuting-triangles-of-maps public open import foundation-core.constant-maps public @@ -59,4 +60,6 @@ open import foundation-core.type-theoretic-principle-of-choice public open import foundation-core.univalence public open import foundation-core.universal-property-pullbacks public open import foundation-core.universal-property-truncation public +open import foundation-core.whiskering-homotopies-concatenation public +open import foundation-core.whiskering-identifications-concatenation public ``` diff --git a/src/foundation-core/coherently-invertible-maps.lagda.md b/src/foundation-core/coherently-invertible-maps.lagda.md index 177ec4b883..596c722080 100644 --- a/src/foundation-core/coherently-invertible-maps.lagda.md +++ b/src/foundation-core/coherently-invertible-maps.lagda.md @@ -7,49 +7,104 @@ module foundation-core.coherently-invertible-maps where
Imports ```agda +open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types +open import foundation.homotopy-algebra open import foundation.universe-levels +open import foundation.whiskering-higher-homotopies-composition open import foundation.whiskering-homotopies-composition +open import foundation-core.commuting-squares-of-homotopies open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.invertible-maps open import foundation-core.retractions open import foundation-core.sections +open import foundation-core.whiskering-homotopies-concatenation ```
## Idea -An [inverse](foundation-core.invertible-maps.md) for a map `f : A → B` is a map -`g : B → A` equipped with [homotopies](foundation-core.homotopies.md) -` (f ∘ g) ~ id` and `(g ∘ f) ~ id`. Such data, however is -[structure](foundation.structure.md) on the map `f`, and not generally a -[property](foundation-core.propositions.md). Therefore we include a coherence -condition for the homotopies of an inverse. A **coherently invertible map** -`f : A → B` is a map equipped with a two-sided inverse and this additional -coherence law. They are also called half-adjoint equivalences. +A [(two-sided) inverse](foundation-core.invertible-maps.md) for a map +`f : A → B` is a map `g : B → A` equipped with +[homotopies](foundation-core.homotopies.md) ` S : f ∘ g ~ id` and +`R : g ∘ f ~ id`. Such data, however is [structure](foundation.structure.md) on +the map `f`, and not generally a [property](foundation-core.propositions.md). +One way to make the type of inverses into a property is by adding a single +coherence condition between the homotopies of the inverse, asking that the +following diagram commmutes + +```text + S ·r f + ---------> + f ∘ g ∘ f f. + ---------> + f ·l R +``` + +We call such data a +{{#concept "coherently invertible map" Agda=coherently-invertible-map}}. I.e., a +coherently invertible map `f : A → B` is a map equipped with a two-sided inverse +and this additional coherence. + +There is also the alternative coherence condition we could add + +```text + R ·r g + ---------> + g ∘ f ∘ g g. + ---------> + g ·l S +``` + +We will colloquially refer to invertible maps equipped with this coherence for +_transpose coherently invertible maps_. + +**Note.** Coherently invertible maps are referred to as +{{#concept "half adjoint equivalences"}} in _Homotopy Type Theory – Univalent +Foundations of Mathematics_. + +On this page we will prove that every two-sided inverse `g` of `f` can be +promoted to a coherent two-sided inverse. Thus, for most properties of +coherently invertible maps, it suffices to consider the data of a two-sided +inverse. However, this coherence construction requires us to replace the section +homotopy (or retraction homotopy). For this reason we also give direct +constructions showing + +1. The identity map is coherently invertible. +2. The composite of two coherently invertible maps is coherently invertible. +3. The inverse of a coherently invertible map is coherently invertible. +4. Every map homotopic to a coherently invertible map is coherently invertible. +5. The 3-for-2 property of coherently invertible maps. + +Each of these constructions appropriately preserve the data of the underlying +two-sided inverse. ## Definition +### The predicate of being a coherently invertible map + ```agda module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} where coherence-is-coherently-invertible : - (f : A → B) (g : B → A) (G : (f ∘ g) ~ id) (H : (g ∘ f) ~ id) → UU (l1 ⊔ l2) - coherence-is-coherently-invertible f g G H = (G ·r f) ~ (f ·l H) + (f : A → B) (g : B → A) (G : f ∘ g ~ id) (H : g ∘ f ~ id) → UU (l1 ⊔ l2) + coherence-is-coherently-invertible f g G H = G ·r f ~ f ·l H is-coherently-invertible : (A → B) → UU (l1 ⊔ l2) is-coherently-invertible f = Σ ( B → A) - ( λ g → Σ ((f ∘ g) ~ id) - ( λ G → Σ ((g ∘ f) ~ id) - ( λ H → coherence-is-coherently-invertible f g G H))) + ( λ g → + Σ ( f ∘ g ~ id) + ( λ G → + Σ ( g ∘ f ~ id) + ( λ H → coherence-is-coherently-invertible f g G H))) module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} @@ -59,118 +114,1285 @@ module _ map-inv-is-coherently-invertible : B → A map-inv-is-coherently-invertible = pr1 H - is-retraction-is-coherently-invertible : - (f ∘ map-inv-is-coherently-invertible) ~ id - is-retraction-is-coherently-invertible = pr1 (pr2 H) + is-section-map-inv-is-coherently-invertible : + is-section f map-inv-is-coherently-invertible + is-section-map-inv-is-coherently-invertible = pr1 (pr2 H) - is-section-is-coherently-invertible : - (map-inv-is-coherently-invertible ∘ f) ~ id - is-section-is-coherently-invertible = pr1 (pr2 (pr2 H)) + is-retraction-map-inv-is-coherently-invertible : + is-retraction f map-inv-is-coherently-invertible + is-retraction-map-inv-is-coherently-invertible = pr1 (pr2 (pr2 H)) coh-is-coherently-invertible : coherence-is-coherently-invertible f ( map-inv-is-coherently-invertible) - ( is-retraction-is-coherently-invertible) - ( is-section-is-coherently-invertible) + ( is-section-map-inv-is-coherently-invertible) + ( is-retraction-map-inv-is-coherently-invertible) coh-is-coherently-invertible = pr2 (pr2 (pr2 H)) is-invertible-is-coherently-invertible : is-invertible f pr1 is-invertible-is-coherently-invertible = map-inv-is-coherently-invertible pr1 (pr2 is-invertible-is-coherently-invertible) = - is-retraction-is-coherently-invertible + is-section-map-inv-is-coherently-invertible pr2 (pr2 is-invertible-is-coherently-invertible) = - is-section-is-coherently-invertible + is-retraction-map-inv-is-coherently-invertible section-is-coherently-invertible : section f - pr1 section-is-coherently-invertible = map-inv-is-coherently-invertible - pr2 section-is-coherently-invertible = is-retraction-is-coherently-invertible + pr1 section-is-coherently-invertible = + map-inv-is-coherently-invertible + pr2 section-is-coherently-invertible = + is-section-map-inv-is-coherently-invertible retraction-is-coherently-invertible : retraction f - pr1 retraction-is-coherently-invertible = map-inv-is-coherently-invertible - pr2 retraction-is-coherently-invertible = is-section-is-coherently-invertible + pr1 retraction-is-coherently-invertible = + map-inv-is-coherently-invertible + pr2 retraction-is-coherently-invertible = + is-retraction-map-inv-is-coherently-invertible +``` + +We will show that `is-coherently-invertible` is a proposition in +[`foundation.coherently-invertible-maps`](foundation.coherently-invertible-maps.md). + +### The type of coherently invertible maps + +```agda +coherently-invertible-map : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) +coherently-invertible-map A B = Σ (A → B) (is-coherently-invertible) + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : coherently-invertible-map A B) + where + + map-coherently-invertible-map : A → B + map-coherently-invertible-map = pr1 e + + is-coherently-invertible-map-coherently-invertible-map : + is-coherently-invertible map-coherently-invertible-map + is-coherently-invertible-map-coherently-invertible-map = pr2 e + + map-inv-coherently-invertible-map : B → A + map-inv-coherently-invertible-map = + map-inv-is-coherently-invertible + ( is-coherently-invertible-map-coherently-invertible-map) + + is-section-map-inv-coherently-invertible-map : + map-coherently-invertible-map ∘ map-inv-coherently-invertible-map ~ id + is-section-map-inv-coherently-invertible-map = + is-section-map-inv-is-coherently-invertible + ( is-coherently-invertible-map-coherently-invertible-map) + + is-retraction-map-inv-coherently-invertible-map : + map-inv-coherently-invertible-map ∘ map-coherently-invertible-map ~ id + is-retraction-map-inv-coherently-invertible-map = + is-retraction-map-inv-is-coherently-invertible + ( is-coherently-invertible-map-coherently-invertible-map) + + coh-coherently-invertible-map : + coherence-is-coherently-invertible + ( map-coherently-invertible-map) + ( map-inv-coherently-invertible-map) + ( is-section-map-inv-coherently-invertible-map) + ( is-retraction-map-inv-coherently-invertible-map) + coh-coherently-invertible-map = + coh-is-coherently-invertible + ( is-coherently-invertible-map-coherently-invertible-map) + + section-coherently-invertible-map : + section map-coherently-invertible-map + section-coherently-invertible-map = + section-is-coherently-invertible + ( is-coherently-invertible-map-coherently-invertible-map) + + retraction-coherently-invertible-map : + retraction map-coherently-invertible-map + retraction-coherently-invertible-map = + retraction-is-coherently-invertible + ( is-coherently-invertible-map-coherently-invertible-map) + + is-invertible-coherently-invertible-map : + is-invertible map-coherently-invertible-map + is-invertible-coherently-invertible-map = + is-invertible-is-coherently-invertible + ( is-coherently-invertible-map-coherently-invertible-map) + + invertible-map-coherently-invertible-map : invertible-map A B + pr1 invertible-map-coherently-invertible-map = + map-coherently-invertible-map + pr2 invertible-map-coherently-invertible-map = + is-invertible-coherently-invertible-map +``` + +### The predicate of being a transpose coherently invertible map + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + coherence-is-transpose-coherently-invertible : + (f : A → B) (g : B → A) (G : f ∘ g ~ id) (H : g ∘ f ~ id) → UU (l1 ⊔ l2) + coherence-is-transpose-coherently-invertible f g G H = H ·r g ~ g ·l G + + is-transpose-coherently-invertible : (A → B) → UU (l1 ⊔ l2) + is-transpose-coherently-invertible f = + Σ ( B → A) + ( λ g → + Σ ( f ∘ g ~ id) + ( λ G → + Σ ( g ∘ f ~ id) + ( λ H → coherence-is-transpose-coherently-invertible f g G H))) + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + (H : is-transpose-coherently-invertible f) + where + + map-inv-is-transpose-coherently-invertible : B → A + map-inv-is-transpose-coherently-invertible = pr1 H + + is-section-map-inv-is-transpose-coherently-invertible : + f ∘ map-inv-is-transpose-coherently-invertible ~ id + is-section-map-inv-is-transpose-coherently-invertible = pr1 (pr2 H) + + is-retraction-map-inv-is-transpose-coherently-invertible : + map-inv-is-transpose-coherently-invertible ∘ f ~ id + is-retraction-map-inv-is-transpose-coherently-invertible = pr1 (pr2 (pr2 H)) + + coh-is-transpose-coherently-invertible : + coherence-is-transpose-coherently-invertible f + ( map-inv-is-transpose-coherently-invertible) + ( is-section-map-inv-is-transpose-coherently-invertible) + ( is-retraction-map-inv-is-transpose-coherently-invertible) + coh-is-transpose-coherently-invertible = pr2 (pr2 (pr2 H)) + + is-invertible-is-transpose-coherently-invertible : is-invertible f + pr1 is-invertible-is-transpose-coherently-invertible = + map-inv-is-transpose-coherently-invertible + pr1 (pr2 is-invertible-is-transpose-coherently-invertible) = + is-section-map-inv-is-transpose-coherently-invertible + pr2 (pr2 is-invertible-is-transpose-coherently-invertible) = + is-retraction-map-inv-is-transpose-coherently-invertible + + section-is-transpose-coherently-invertible : section f + pr1 section-is-transpose-coherently-invertible = + map-inv-is-transpose-coherently-invertible + pr2 section-is-transpose-coherently-invertible = + is-section-map-inv-is-transpose-coherently-invertible + + retraction-is-transpose-coherently-invertible : retraction f + pr1 retraction-is-transpose-coherently-invertible = + map-inv-is-transpose-coherently-invertible + pr2 retraction-is-transpose-coherently-invertible = + is-retraction-map-inv-is-transpose-coherently-invertible +``` + +We will show that `is-transpose-coherently-invertible` is a proposition in +[`foundation.coherently-invertible-maps`](foundation.coherently-invertible-maps.md). + +### The type of transpose coherently invertible maps + +```agda +transpose-coherently-invertible-map : + {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) +transpose-coherently-invertible-map A B = + Σ (A → B) (is-transpose-coherently-invertible) + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + (e : transpose-coherently-invertible-map A B) + where + + map-transpose-coherently-invertible-map : A → B + map-transpose-coherently-invertible-map = pr1 e + + is-transpose-coherently-invertible-map-transpose-coherently-invertible-map : + is-transpose-coherently-invertible map-transpose-coherently-invertible-map + is-transpose-coherently-invertible-map-transpose-coherently-invertible-map = + pr2 e + + map-inv-transpose-coherently-invertible-map : B → A + map-inv-transpose-coherently-invertible-map = + map-inv-is-transpose-coherently-invertible + ( is-transpose-coherently-invertible-map-transpose-coherently-invertible-map) + + is-section-map-inv-transpose-coherently-invertible-map : + ( map-transpose-coherently-invertible-map ∘ + map-inv-transpose-coherently-invertible-map) ~ + ( id) + is-section-map-inv-transpose-coherently-invertible-map = + is-section-map-inv-is-transpose-coherently-invertible + ( is-transpose-coherently-invertible-map-transpose-coherently-invertible-map) + + is-retraction-map-inv-transpose-coherently-invertible-map : + ( map-inv-transpose-coherently-invertible-map ∘ + map-transpose-coherently-invertible-map) ~ + ( id) + is-retraction-map-inv-transpose-coherently-invertible-map = + is-retraction-map-inv-is-transpose-coherently-invertible + ( is-transpose-coherently-invertible-map-transpose-coherently-invertible-map) + + coh-transpose-coherently-invertible-map : + coherence-is-transpose-coherently-invertible + ( map-transpose-coherently-invertible-map) + ( map-inv-transpose-coherently-invertible-map) + ( is-section-map-inv-transpose-coherently-invertible-map) + ( is-retraction-map-inv-transpose-coherently-invertible-map) + coh-transpose-coherently-invertible-map = + coh-is-transpose-coherently-invertible + ( is-transpose-coherently-invertible-map-transpose-coherently-invertible-map) + + section-transpose-coherently-invertible-map : + section map-transpose-coherently-invertible-map + section-transpose-coherently-invertible-map = + section-is-transpose-coherently-invertible + ( is-transpose-coherently-invertible-map-transpose-coherently-invertible-map) + + retraction-transpose-coherently-invertible-map : + retraction map-transpose-coherently-invertible-map + retraction-transpose-coherently-invertible-map = + retraction-is-transpose-coherently-invertible + ( is-transpose-coherently-invertible-map-transpose-coherently-invertible-map) + + is-invertible-transpose-coherently-invertible-map : + is-invertible map-transpose-coherently-invertible-map + is-invertible-transpose-coherently-invertible-map = + is-invertible-is-transpose-coherently-invertible + ( is-transpose-coherently-invertible-map-transpose-coherently-invertible-map) + + invertible-map-transpose-coherently-invertible-map : invertible-map A B + pr1 invertible-map-transpose-coherently-invertible-map = + map-transpose-coherently-invertible-map + pr2 invertible-map-transpose-coherently-invertible-map = + is-invertible-transpose-coherently-invertible-map +``` + +### Invertible maps that are coherent are coherently invertible maps + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + (H : is-invertible f) + where + + coherence-is-invertible : UU (l1 ⊔ l2) + coherence-is-invertible = + coherence-is-coherently-invertible + ( f) + ( map-inv-is-invertible H) + ( is-section-map-inv-is-invertible H) + ( is-retraction-map-inv-is-invertible H) + + transpose-coherence-is-invertible : UU (l1 ⊔ l2) + transpose-coherence-is-invertible = + coherence-is-transpose-coherently-invertible + ( f) + ( map-inv-is-invertible H) + ( is-section-map-inv-is-invertible H) + ( is-retraction-map-inv-is-invertible H) + + is-coherently-invertible-coherence-is-invertible : + coherence-is-invertible → is-coherently-invertible f + is-coherently-invertible-coherence-is-invertible coh = + ( map-inv-is-invertible H , + is-section-map-inv-is-invertible H , + is-retraction-map-inv-is-invertible H , + coh) + + is-transpose-coherently-invertible-transpose-coherence-is-invertible : + transpose-coherence-is-invertible → is-transpose-coherently-invertible f + is-transpose-coherently-invertible-transpose-coherence-is-invertible coh = + ( map-inv-is-invertible H , + is-section-map-inv-is-invertible H , + is-retraction-map-inv-is-invertible H , + coh) ``` ## Properties +### The inverse of a coherently invertible map is transpose coherently invertible and vice versa + +The inverse of a coherently invertible map is transpose coherently invertible. +Conversely, the inverse of a transpose coherently invertible map is coherently +invertible. Since these are defined by simply moving data around, they are +strict inverses to one another. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-transpose-coherently-invertible-map-inv-is-coherently-invertible : + {f : A → B} (H : is-coherently-invertible f) → + is-transpose-coherently-invertible (map-inv-is-coherently-invertible H) + is-transpose-coherently-invertible-map-inv-is-coherently-invertible {f} H = + ( f , + is-retraction-map-inv-is-coherently-invertible H , + is-section-map-inv-is-coherently-invertible H , + coh-is-coherently-invertible H) + + is-coherently-invertible-map-inv-is-transpose-coherently-invertible : + {f : A → B} (H : is-transpose-coherently-invertible f) → + is-coherently-invertible (map-inv-is-transpose-coherently-invertible H) + is-coherently-invertible-map-inv-is-transpose-coherently-invertible {f} H = + ( f , + is-retraction-map-inv-is-transpose-coherently-invertible H , + is-section-map-inv-is-transpose-coherently-invertible H , + coh-is-transpose-coherently-invertible H) + + transpose-coherently-invertible-map-inv-coherently-invertible-map : + coherently-invertible-map A B → transpose-coherently-invertible-map B A + transpose-coherently-invertible-map-inv-coherently-invertible-map e = + ( map-inv-coherently-invertible-map e , + is-transpose-coherently-invertible-map-inv-is-coherently-invertible + ( is-coherently-invertible-map-coherently-invertible-map e)) + + coherently-invertible-map-inv-transpose-coherently-invertible-map : + transpose-coherently-invertible-map A B → coherently-invertible-map B A + coherently-invertible-map-inv-transpose-coherently-invertible-map e = + ( map-inv-transpose-coherently-invertible-map e , + is-coherently-invertible-map-inv-is-transpose-coherently-invertible + ( is-transpose-coherently-invertible-map-transpose-coherently-invertible-map + ( e))) +``` + ### Invertible maps are coherently invertible -#### Lemma: A coherence for homotopies to an identity map +This result is known as +[Vogt's lemma](https://ncatlab.org/nlab/show/homotopy+equivalence#vogts_lemma) +in point-set topology. The construction follows Lemma 10.4.5 in _Introduction to +Homotopy Type Theory_. ```agda -coh-is-coherently-invertible-id : - {l : Level} {A : UU l} {f : A → A} (H : f ~ (λ x → x)) → - (x : A) → H (f x) ＝ ap f (H x) -coh-is-coherently-invertible-id {_} {A} {f} H x = - is-injective-concat' (H x) - ( ( ap (concat (H (f x)) x) (inv (ap-id (H x)))) ∙ - ( nat-htpy H (H x))) +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} (H : is-invertible f) + where + + is-retraction-map-inv-is-coherently-invertible-is-invertible : + map-inv-is-invertible H ∘ f ~ id + is-retraction-map-inv-is-coherently-invertible-is-invertible = + is-retraction-map-inv-is-invertible H + + abstract + is-section-map-inv-is-coherently-invertible-is-invertible : + f ∘ map-inv-is-invertible H ~ id + is-section-map-inv-is-coherently-invertible-is-invertible = + ( ( inv-htpy (is-section-map-inv-is-invertible H)) ·r + ( f ∘ map-inv-is-invertible H)) ∙h + ( ( ( f) ·l + ( is-retraction-map-inv-is-invertible H) ·r + ( map-inv-is-invertible H)) ∙h + ( is-section-map-inv-is-invertible H)) + + abstract + inv-coh-is-coherently-invertible-is-invertible : + f ·l is-retraction-map-inv-is-coherently-invertible-is-invertible ~ + is-section-map-inv-is-coherently-invertible-is-invertible ·r f + inv-coh-is-coherently-invertible-is-invertible = + left-transpose-htpy-concat + ( ( is-section-map-inv-is-invertible H) ·r + ( f ∘ map-inv-is-invertible H ∘ f)) + ( f ·l is-retraction-map-inv-is-invertible H) + ( ( ( f) ·l + ( is-retraction-map-inv-is-invertible H) ·r + ( map-inv-is-invertible H ∘ f)) ∙h + ( is-section-map-inv-is-invertible H ·r f)) + ( ( ( nat-htpy (is-section-map-inv-is-invertible H ·r f)) ·r + ( is-retraction-map-inv-is-invertible H)) ∙h + ( right-whisker-concat-htpy + ( ( inv-preserves-comp-left-whisker-comp + ( f) + ( map-inv-is-invertible H ∘ f) + ( is-retraction-map-inv-is-invertible H)) ∙h + ( left-whisker-comp² + ( f) + ( inv-coh-htpy-id (is-retraction-map-inv-is-invertible H)))) + ( is-section-map-inv-is-invertible H ·r f))) + + abstract + coh-is-coherently-invertible-is-invertible : + coherence-is-coherently-invertible + ( f) + ( map-inv-is-invertible H) + ( is-section-map-inv-is-coherently-invertible-is-invertible) + ( is-retraction-map-inv-is-coherently-invertible-is-invertible) + coh-is-coherently-invertible-is-invertible = + inv-htpy inv-coh-is-coherently-invertible-is-invertible + + is-coherently-invertible-is-invertible : is-coherently-invertible f + is-coherently-invertible-is-invertible = + ( map-inv-is-invertible H , + is-section-map-inv-is-coherently-invertible-is-invertible , + is-retraction-map-inv-is-coherently-invertible-is-invertible , + coh-is-coherently-invertible-is-invertible) ``` -#### The proof that invertible maps are coherently invertible +We get the transpose version for free: + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} (H : is-invertible f) + where + + is-transpose-coherently-invertible-is-invertible : + is-transpose-coherently-invertible f + is-transpose-coherently-invertible-is-invertible = + is-transpose-coherently-invertible-map-inv-is-coherently-invertible + ( is-coherently-invertible-is-invertible + ( is-invertible-map-inv-is-invertible H)) +``` + +### Coherently invertible maps are injective + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + where + + is-injective-is-coherently-invertible : + (H : is-coherently-invertible f) {x y : A} → f x ＝ f y → x ＝ y + is-injective-is-coherently-invertible H = + is-injective-is-invertible + ( is-invertible-is-coherently-invertible H) + + is-injective-is-transpose-coherently-invertible : + (H : is-transpose-coherently-invertible f) {x y : A} → f x ＝ f y → x ＝ y + is-injective-is-transpose-coherently-invertible H = + is-injective-is-invertible + ( is-invertible-is-transpose-coherently-invertible H) +``` + +### Coherently invertible maps are embeddings + +We show that `is-injective-is-coherently-invertible` is a +[section](foundation-core.sections.md) and +[retraction](foundation-core.retractions.md) of `ap f`. ```agda module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + (H : is-coherently-invertible f) {x y : A} where abstract - is-section-map-inv-is-invertible : - (H : is-invertible f) → (f ∘ map-inv-is-invertible H) ~ id - is-section-map-inv-is-invertible H y = - ( inv (is-retraction-is-invertible H (f (map-inv-is-invertible H y)))) ∙ - ( ( ap f (is-section-is-invertible H (map-inv-is-invertible H y))) ∙ - ( is-retraction-is-invertible H y)) - - is-retraction-map-inv-is-invertible : - (H : is-invertible f) → (map-inv-is-invertible H ∘ f) ~ id - is-retraction-map-inv-is-invertible = is-section-is-invertible - - inv-coherence-map-inv-is-invertible : - (H : is-invertible f) → - f ·l is-retraction-map-inv-is-invertible H ~ - is-section-map-inv-is-invertible H ·r f - inv-coherence-map-inv-is-invertible H x = - left-transpose-eq-concat - ( is-retraction-is-invertible H (f (map-inv-is-invertible H (f x)))) - ( ap f (is-section-is-invertible H x)) - ( ( ap f - ( is-section-is-invertible H (map-inv-is-invertible H (f x)))) ∙ - ( is-retraction-is-invertible H (f x))) - ( ( nat-htpy - ( right-whisker-comp (is-retraction-is-invertible H) f) - ( is-section-is-invertible H x)) ∙ - ( ap - ( concat' _ (is-retraction-is-invertible H (f x))) - ( ( ap-comp f - ( map-inv-is-invertible H ∘ f) - ( is-section-is-invertible H x)) ∙ - ( inv - ( ap - ( ap f) - ( coh-is-coherently-invertible-id - ( is-section-is-invertible H) x)))))) - - coherence-map-inv-is-invertible : - ( H : is-invertible f) → - ( is-section-map-inv-is-invertible H ·r f) ~ - ( f ·l is-retraction-map-inv-is-invertible H) - coherence-map-inv-is-invertible H x = - inv (inv-coherence-map-inv-is-invertible H x) + is-section-is-injective-is-coherently-invertible : + ap f {x} {y} ∘ is-injective-is-coherently-invertible H ~ id + is-section-is-injective-is-coherently-invertible p = + ( ap-concat f + ( inv (is-retraction-map-inv-is-coherently-invertible H x)) + ( ( ap (map-inv-is-coherently-invertible H) p) ∙ + ( is-retraction-map-inv-is-coherently-invertible H y))) ∙ + ( ap-binary + ( _∙_) + ( ap-inv f (is-retraction-map-inv-is-coherently-invertible H x)) + ( ( ap-concat f + ( ap (map-inv-is-coherently-invertible H) p) + ( is-retraction-map-inv-is-coherently-invertible H y)) ∙ + ( ap-binary + ( _∙_) + ( inv (ap-comp f (map-inv-is-coherently-invertible H) p)) + ( inv (coh-is-coherently-invertible H y))))) ∙ + ( inv + ( left-transpose-eq-concat + ( ap f (is-retraction-map-inv-is-coherently-invertible H x)) + ( p) + ( ( ap (f ∘ map-inv-is-coherently-invertible H) p) ∙ + ( is-section-map-inv-is-coherently-invertible H (f y))) + ( ( ap-binary + ( _∙_) + ( inv (coh-is-coherently-invertible H x)) + ( inv (ap-id p))) ∙ + ( nat-htpy (is-section-map-inv-is-coherently-invertible H) p)))) + + abstract + is-retraction-is-injective-is-coherently-invertible : + is-injective-is-coherently-invertible H ∘ ap f {x} {y} ~ id + is-retraction-is-injective-is-coherently-invertible refl = + left-inv (is-retraction-map-inv-is-coherently-invertible H x) + + is-invertible-ap-is-coherently-invertible : is-invertible (ap f {x} {y}) + is-invertible-ap-is-coherently-invertible = + ( is-injective-is-coherently-invertible H , + is-section-is-injective-is-coherently-invertible , + is-retraction-is-injective-is-coherently-invertible) + + is-coherently-invertible-ap-is-coherently-invertible : + is-coherently-invertible (ap f {x} {y}) + is-coherently-invertible-ap-is-coherently-invertible = + is-coherently-invertible-is-invertible + ( is-invertible-ap-is-coherently-invertible) +``` + +### Coherently invertible maps are transpose coherently invertible + +The proof follows Lemma 4.2.2 in _Homotopy Type Theory – Univalent Foundations +of Mathematics_. + +**Proof.** By naturality of homotopies we have + +```text + gfRg + gfgfg --------------------> gfg + | | + Rgfg | nat-htpy R ·r (R ·r g) | Rg + ∨ ∨ + gfg ----------------------> g. + Rg +``` + +We can paste the homotopy + +```text + g (inv-coh-htpy-id S) ∙ gHg : Sgfg ~ gfSg ~ gfRg +``` + +along the top edge of this naturality square obtaining the coherence square + +```text + gfgS + gfgfg -------> gfg + | | + Rgfg | | Rg + ∨ ∨ + gfg ---------> g. + Rg +``` + +There is also the naturality square + +```text + gfgS + gfgfg --------------------> gfg + | | + Rgfg | nat-htpy (R ·r g) ·r S | Rg + ∨ ∨ + gfg ----------------------> g. + gS +``` + +Now, by pasting these along the common edge `Rgfg`, we obtain + +```text + gfgS gfgS + gfg <------- gfgfg -------> gfg + | | | + Rg | | Rgfg | Rg + ∨ ∨ ∨ + g <--------- gfg --------> gm + Rg gS +``` + +After cancelling `gfgS` and `Rg` with themselves, we are left with `Rg ~ gS` as +desired. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + (f : A → B) + (g : B → A) + (S : is-section f g) + (R : is-retraction f g) + (H : coherence-is-coherently-invertible f g S R) + where + + abstract + coh-transposition-coherence-is-coherently-invertible : + (g ∘ f ∘ g) ·l S ~ (g ∘ f) ·l R ·r g + coh-transposition-coherence-is-coherently-invertible = + ( inv-preserves-comp-left-whisker-comp g (f ∘ g) S) ∙h + ( left-whisker-comp² g (inv-coh-htpy-id S)) ∙h + ( double-whisker-comp² g H g) ∙h + ( preserves-comp-left-whisker-comp g f (R ·r g)) + + abstract + naturality-square-transposition-coherence-is-coherently-invertible : + coherence-square-homotopies + ( R ·r (g ∘ f ∘ g)) + ( (g ∘ f ∘ g) ·l S) + ( R ·r g) + ( R ·r g) + naturality-square-transposition-coherence-is-coherently-invertible = + ( ap-concat-htpy' + ( R ·r g) + ( coh-transposition-coherence-is-coherently-invertible)) ∙h + ( inv-htpy (nat-htpy R ·r (R ·r g))) ∙h + ( ap-concat-htpy + ( R ·r (g ∘ f ∘ g)) + ( left-unit-law-left-whisker-comp (R ·r g))) + + abstract + coherence-transposition-coherence-is-coherently-invertible : + coherence-is-transpose-coherently-invertible f g S R + coherence-transposition-coherence-is-coherently-invertible = + ( ap-concat-htpy' + ( R ·r g) + ( inv-htpy (left-inv-htpy ((g ∘ f ∘ g) ·l S)))) ∙h + ( assoc-htpy (inv-htpy ((g ∘ f ∘ g) ·l S)) ((g ∘ f ∘ g) ·l S) (R ·r g)) ∙h + ( ap-concat-htpy + ( inv-htpy ((g ∘ f ∘ g) ·l S)) + ( inv-htpy (nat-htpy (R ·r g) ·r S))) ∙h + ( inv-htpy + ( assoc-htpy + ( inv-htpy ((g ∘ f ∘ g) ·l S)) + ( R ·r (g ∘ f ∘ g)) + ( g ·l S))) ∙h + ( ap-concat-htpy' + ( g ·l S) + ( ( vertical-inv-coherence-square-homotopies + ( R ·r (g ∘ f ∘ g)) ((g ∘ f ∘ g) ·l S) (R ·r g) (R ·r g) + ( naturality-square-transposition-coherence-is-coherently-invertible)) ∙h + ( right-inv-htpy (R ·r g)))) +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + where + + coherence-transposition-is-coherently-invertible : + (H : is-coherently-invertible f) → + coherence-is-transpose-coherently-invertible + ( f) + ( map-inv-is-coherently-invertible H) + ( is-section-map-inv-is-coherently-invertible H) + ( is-retraction-map-inv-is-coherently-invertible H) + coherence-transposition-is-coherently-invertible + ( g , S , R , H) = + coherence-transposition-coherence-is-coherently-invertible f g S R H + + transposition-is-coherently-invertible : + is-coherently-invertible f → is-transpose-coherently-invertible f + transposition-is-coherently-invertible H = + is-transpose-coherently-invertible-transpose-coherence-is-invertible + ( is-invertible-is-coherently-invertible H) + ( coherence-transposition-is-coherently-invertible H) +``` + +### Transpose coherently invertible maps are coherently invertible + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + where + + coherence-transposition-is-transpose-coherently-invertible : + (H : is-transpose-coherently-invertible f) → + coherence-is-coherently-invertible + ( f) + ( map-inv-is-transpose-coherently-invertible H) + ( is-section-map-inv-is-transpose-coherently-invertible H) + ( is-retraction-map-inv-is-transpose-coherently-invertible H) + coherence-transposition-is-transpose-coherently-invertible H = + coherence-transposition-is-coherently-invertible + ( is-coherently-invertible-map-inv-is-transpose-coherently-invertible H) + + transposition-is-transpose-coherently-invertible : + is-transpose-coherently-invertible f → is-coherently-invertible f + transposition-is-transpose-coherently-invertible H = + is-coherently-invertible-coherence-is-invertible + ( is-invertible-is-transpose-coherently-invertible H) + ( coherence-transposition-is-transpose-coherently-invertible H) + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + transposition-coherently-invertible-map : + coherently-invertible-map A B → transpose-coherently-invertible-map A B + transposition-coherently-invertible-map (f , H) = + ( f , transposition-is-coherently-invertible H) + + transposition-transpose-coherently-invertible-map : + transpose-coherently-invertible-map A B → coherently-invertible-map A B + transposition-transpose-coherently-invertible-map (f , H) = + ( f , transposition-is-transpose-coherently-invertible H) +``` + +### Coherently invertible maps are closed under homotopies + +**Proof.** Assume given a coherently invertible map `f` with inverse `g`, +homotopies `S : f ∘ g ~ id`, `R : g ∘ f ~ id` and coherence `C : Sf ~ fR`. +Moreover, assume the map `f'` is homotopic to `f` with homotopy `H : f' ~ f`. +Then `g` is also a two-sided inverse to `f'` via the homotopies + +```text + S' := Hg ∙ S : f' ∘ g ~ id and R' := gH ∙ R : g ∘ f' ~ id. +``` + +Moreover, these witnesses are part of a coherent inverse to `f'`. To show this, +we must construct a coherence `C'` of the square + +```text + Hgf' + f'gf' -----> f'gf + | | + f'gH | | Sf' + ∨ ∨ + f'gf -------> f'. + f'R +``` + +We begin by observing that `C` fits somewhere along the diagonal of this square +via the composite + +```text + Sf + HgH ------> H⁻¹ + f'gf' -----> fgf C f ----> f'. + ------> + fR +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f f' : A → B} (H : f' ~ f) + (g : B → A) (S : f ∘ g ~ id) (R : g ∘ f ~ id) (C : S ·r f ~ f ·l R) + where + + coh-coh-is-coherently-invertible-htpy : + horizontal-concat-htpy' (H ·r g) H ∙h (S ·r f ∙h inv-htpy H) ~ + horizontal-concat-htpy' (H ·r g) H ∙h (f ·l R ∙h inv-htpy H) + coh-coh-is-coherently-invertible-htpy = + left-whisker-concat-htpy + ( horizontal-concat-htpy' (H ·r g) H) + ( right-whisker-concat-htpy C (inv-htpy H)) +``` + +Now the problem is reduced to constructing two homotopies + +```text + Hgf' ∙ Sf' ~ HgH ∙ Sf ∙ H⁻¹ and f'gH ∙ f'R ~ HgH ∙ fR ∙ H⁻¹. +``` + +By the two equivalent constructions of the horizontal composite `HgH` + +```text + Hgf' ∙ fgH ~ HgH ~ f'gH ∙ Hgf +``` + +constructing the two homotopies is equivalent to constructing coherences of the +squares + +```text + fgH Hgf + fgf' -------> fgf f'gf -------> fgf + | | | | + Sf' | | Sf f'R | | fR + ∨ ∨ ∨ ∨ + f' ---------> f f' ---------> f + H H +``` + +However, these squares are naturality squares, so we are done. + +```agda + coh-section-is-coherently-invertible-htpy : + (H ·r g ∙h S) ·r f' ~ + horizontal-concat-htpy' (H ·r g) H ∙h ((S ·r f) ∙h inv-htpy H) + coh-section-is-coherently-invertible-htpy = + ( left-whisker-concat-htpy + ( H ·r (g ∘ f')) + ( right-transpose-htpy-concat (S ·r f') H ((f ∘ g) ·l H ∙h S ·r f) + ( ( ap-concat-htpy + ( S ·r f') + ( inv-htpy (left-unit-law-left-whisker-comp H))) ∙h + ( nat-htpy S ·r H)))) ∙h + ( ( ap-concat-htpy + ( H ·r (g ∘ f')) + ( assoc-htpy ((f ∘ g) ·l H) (S ·r f) (inv-htpy H))) ∙h + ( inv-htpy + ( assoc-htpy (H ·r (g ∘ f')) ((f ∘ g) ·l H) ((S ·r f) ∙h inv-htpy H)))) + + coh-retraction-is-coherently-invertible-htpy : + horizontal-concat-htpy' (H ·r g) H ∙h ((f ·l R) ∙h inv-htpy H) ~ + f' ·l ((g ·l H) ∙h R) + coh-retraction-is-coherently-invertible-htpy = + ( ap-concat-htpy' + ( f ·l R ∙h inv-htpy H) + ( coh-horizontal-concat-htpy (H ·r g) H)) ∙h + ( assoc-htpy ((f' ∘ g) ·l H) (H ·r (g ∘ f)) (f ·l R ∙h inv-htpy H)) ∙h + ( ap-concat-htpy + ( (f' ∘ g) ·l H) + ( inv-htpy (assoc-htpy (H ·r (g ∘ f)) (f ·l R) (inv-htpy H)))) ∙h + ( left-whisker-concat-htpy + ( (f' ∘ g) ·l H) + ( inv-htpy + ( right-transpose-htpy-concat (f' ·l R) H ((H ·r (g ∘ f) ∙h f ·l R)) + ( inv-htpy (nat-htpy H ·r R))))) ∙h + ( ap-concat-htpy' + ( f' ·l R) + ( inv-preserves-comp-left-whisker-comp f' g H)) ∙h + ( inv-htpy (distributive-left-whisker-comp-concat f' (g ·l H) R)) +``` + +Finally, we concatenate the three coherences to obtain our desired result. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f f' : A → B} + where + + abstract + coh-is-coherently-invertible-htpy : + (H : f' ~ f) (F : is-coherently-invertible f) → + coherence-is-coherently-invertible + ( f') + ( map-inv-is-coherently-invertible F) + ( is-section-map-inv-is-invertible-htpy + ( H) + ( is-invertible-is-coherently-invertible F)) + ( is-retraction-map-inv-is-invertible-htpy + ( H) + ( is-invertible-is-coherently-invertible F)) + coh-is-coherently-invertible-htpy H F = + ( coh-section-is-coherently-invertible-htpy H + ( map-inv-is-coherently-invertible F) + ( is-section-map-inv-is-coherently-invertible F) + ( is-retraction-map-inv-is-coherently-invertible F) + ( coh-is-coherently-invertible F)) ∙h + ( coh-coh-is-coherently-invertible-htpy H + ( map-inv-is-coherently-invertible F) + ( is-section-map-inv-is-coherently-invertible F) + ( is-retraction-map-inv-is-coherently-invertible F) + ( coh-is-coherently-invertible F)) ∙h + ( coh-retraction-is-coherently-invertible-htpy H + ( map-inv-is-coherently-invertible F) + ( is-section-map-inv-is-coherently-invertible F) + ( is-retraction-map-inv-is-coherently-invertible F) + ( coh-is-coherently-invertible F)) + + is-coherently-invertible-htpy : + f' ~ f → is-coherently-invertible f → is-coherently-invertible f' + is-coherently-invertible-htpy H F = + is-coherently-invertible-coherence-is-invertible + ( is-invertible-htpy H (is-invertible-is-coherently-invertible F)) + ( coh-is-coherently-invertible-htpy H F) + + is-coherently-invertible-inv-htpy : + f ~ f' → is-coherently-invertible f → is-coherently-invertible f' + is-coherently-invertible-inv-htpy H = + is-coherently-invertible-htpy (inv-htpy H) + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + where + + is-coherently-invertible-htpy-coherently-invertible-map : + (e : coherently-invertible-map A B) → + f ~ map-coherently-invertible-map e → + is-coherently-invertible f + is-coherently-invertible-htpy-coherently-invertible-map (e , E) H = + is-coherently-invertible-htpy H E + + is-coherently-invertible-inv-htpy-coherently-invertible-map : + (e : coherently-invertible-map A B) → + map-coherently-invertible-map e ~ f → + is-coherently-invertible f + is-coherently-invertible-inv-htpy-coherently-invertible-map (e , E) H = + is-coherently-invertible-inv-htpy H E +``` + +### The identity map is coherently invertible + +```agda +module _ + {l : Level} {A : UU l} + where + + is-coherently-invertible-id : + is-coherently-invertible (id {A = A}) + is-coherently-invertible-id = + is-coherently-invertible-coherence-is-invertible is-invertible-id refl-htpy + + id-coherently-invertible-map : coherently-invertible-map A A + id-coherently-invertible-map = + ( id , is-coherently-invertible-id) + + is-transpose-coherently-invertible-id : + is-transpose-coherently-invertible (id {A = A}) + is-transpose-coherently-invertible-id = + is-transpose-coherently-invertible-map-inv-is-coherently-invertible + ( is-coherently-invertible-id) + + id-transpose-coherently-invertible-map : + transpose-coherently-invertible-map A A + id-transpose-coherently-invertible-map = + ( id , is-transpose-coherently-invertible-id) +``` + +### Inversion of coherently invertible maps + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-coherently-invertible-map-inv-is-coherently-invertible : + {f : A → B} (H : is-coherently-invertible f) → + is-coherently-invertible (map-inv-is-coherently-invertible H) + is-coherently-invertible-map-inv-is-coherently-invertible H = + is-coherently-invertible-map-inv-is-transpose-coherently-invertible + ( transposition-is-coherently-invertible H) + + is-transpose-coherently-invertible-map-inv-is-transpose-coherently-invertible : + {f : A → B} (H : is-transpose-coherently-invertible f) → + is-transpose-coherently-invertible + ( map-inv-is-transpose-coherently-invertible H) + is-transpose-coherently-invertible-map-inv-is-transpose-coherently-invertible + H = + transposition-is-coherently-invertible + ( is-coherently-invertible-map-inv-is-transpose-coherently-invertible H) + + inv-coherently-invertible-map : + coherently-invertible-map A B → coherently-invertible-map B A + inv-coherently-invertible-map (f , H) = + ( map-inv-is-coherently-invertible H , + is-coherently-invertible-map-inv-is-coherently-invertible H) + + inv-transpose-coherently-invertible-map : + transpose-coherently-invertible-map A B → + transpose-coherently-invertible-map B A + inv-transpose-coherently-invertible-map (f , H) = + ( map-inv-is-transpose-coherently-invertible H , + is-transpose-coherently-invertible-map-inv-is-transpose-coherently-invertible + ( H)) +``` + +### Composition of coherently invertible maps + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + (g : B → C) (f : A → B) + (G : is-coherently-invertible g) + (F : is-coherently-invertible f) + where + + coh-is-coherently-invertible-comp : + coherence-is-coherently-invertible + ( g ∘ f) + ( map-inv-is-invertible-comp g f + ( is-invertible-is-coherently-invertible G) + ( is-invertible-is-coherently-invertible F)) + ( is-section-map-inv-is-invertible-comp g f + ( is-invertible-is-coherently-invertible G) + ( is-invertible-is-coherently-invertible F)) + ( is-retraction-map-inv-is-invertible-comp g f + ( is-invertible-is-coherently-invertible G) + ( is-invertible-is-coherently-invertible F)) + coh-is-coherently-invertible-comp = + ( ap-concat-htpy + ( ( g) ·l + ( is-section-map-inv-is-coherently-invertible F) ·r + ( map-inv-is-coherently-invertible G ∘ g ∘ f)) + ( coh-is-coherently-invertible G ·r f)) ∙h + ( right-whisker-comp² + ( ( nat-htpy (g ·l is-section-map-inv-is-coherently-invertible F)) ·r + ( is-retraction-map-inv-is-coherently-invertible G)) + ( f)) ∙h + ( ap-binary-concat-htpy + ( inv-preserves-comp-left-whisker-comp + ( g ∘ f) + ( map-inv-is-coherently-invertible F) + ( is-retraction-map-inv-is-coherently-invertible G ·r f)) + ( ( left-whisker-comp² g (coh-is-coherently-invertible F)) ∙h + ( preserves-comp-left-whisker-comp g f + ( is-retraction-map-inv-is-coherently-invertible F)))) ∙h + ( inv-htpy + ( distributive-left-whisker-comp-concat + ( g ∘ f) + ( ( map-inv-is-coherently-invertible F) ·l + ( is-retraction-map-inv-is-coherently-invertible G ·r f)) + ( is-retraction-map-inv-is-coherently-invertible F))) + + is-coherently-invertible-comp : is-coherently-invertible (g ∘ f) + is-coherently-invertible-comp = + is-coherently-invertible-coherence-is-invertible + ( is-invertible-comp g f + ( is-invertible-is-coherently-invertible G) + ( is-invertible-is-coherently-invertible F)) + ( coh-is-coherently-invertible-comp) + +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + (g : B → C) (f : A → B) + (G : is-transpose-coherently-invertible g) + (F : is-transpose-coherently-invertible f) + where + + coh-is-transpose-coherently-invertible-comp : + coherence-is-transpose-coherently-invertible + ( g ∘ f) + ( map-inv-is-invertible-comp g f + ( is-invertible-is-transpose-coherently-invertible G) + ( is-invertible-is-transpose-coherently-invertible F)) + ( is-section-map-inv-is-invertible-comp g f + ( is-invertible-is-transpose-coherently-invertible G) + ( is-invertible-is-transpose-coherently-invertible F)) + ( is-retraction-map-inv-is-invertible-comp g f + ( is-invertible-is-transpose-coherently-invertible G) + ( is-invertible-is-transpose-coherently-invertible F)) + coh-is-transpose-coherently-invertible-comp = + coh-is-coherently-invertible-comp + ( map-inv-is-transpose-coherently-invertible F) + ( map-inv-is-transpose-coherently-invertible G) + ( is-coherently-invertible-map-inv-is-transpose-coherently-invertible F) + ( is-coherently-invertible-map-inv-is-transpose-coherently-invertible G) + + is-transpose-coherently-invertible-comp : + is-transpose-coherently-invertible (g ∘ f) + is-transpose-coherently-invertible-comp = + is-transpose-coherently-invertible-transpose-coherence-is-invertible + ( is-invertible-comp g f + ( is-invertible-is-transpose-coherently-invertible G) + ( is-invertible-is-transpose-coherently-invertible F)) + ( coh-is-transpose-coherently-invertible-comp) + +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + where + + comp-coherently-invertible-map : + coherently-invertible-map B C → + coherently-invertible-map A B → + coherently-invertible-map A C + comp-coherently-invertible-map (g , G) (f , F) = + ( g ∘ f , is-coherently-invertible-comp g f G F) + + comp-transpose-coherently-invertible-map : + transpose-coherently-invertible-map B C → + transpose-coherently-invertible-map A B → + transpose-coherently-invertible-map A C + comp-transpose-coherently-invertible-map (g , G) (f , F) = + ( g ∘ f , is-transpose-coherently-invertible-comp g f G F) +``` + +### The 3-for-2 property of coherently invertible maps + +The +{{#concept "3-for-2 property" Disambiguation="of coherently invertible maps"}} +of coherently invertible maps asserts that for any +[commuting triangle](foundation-core.commuting-triangles-of-maps.md) of maps + +```text + h + A ------> B + \ / + f\ /g + \ / + V V + X, +``` + +if two of the three maps are coherently invertible, then so is the third. + +We also record special cases of the 3-for-2 property of coherently invertible +maps, where we only assume maps `g : B → X` and `h : A → B`. In this special +case, we set `f := g ∘ h` and the triangle commutes by `refl-htpy`. + +[André Joyal](https://en.wikipedia.org/wiki/André_Joyal) proposed calling this +property the 3-for-2 property, despite most mathematicians calling it the +_2-out-of-3 property_. The story goes that on the produce market is is common to +advertise a discount as "3-for-2". If you buy two apples, then you get the third +for free. Similarly, if you prove that two maps in a commuting triangle are +coherently invertible, then you get the third proof for free. + +#### The left map in a commuting triangle is coherently invertible if the other two maps are + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + (f : A → X) (g : B → X) (h : A → B) (T : f ~ g ∘ h) + where + + is-coherently-invertible-left-map-triangle : + is-coherently-invertible h → + is-coherently-invertible g → + is-coherently-invertible f + is-coherently-invertible-left-map-triangle H G = + is-coherently-invertible-htpy T (is-coherently-invertible-comp g h G H) +``` + +#### The right map in a commuting triangle is coherently invertible if the other two maps are + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + (f : A → X) (g : B → X) (h : A → B) (T : f ~ g ∘ h) + where + + is-coherently-invertible-right-map-triangle : + is-coherently-invertible f → + is-coherently-invertible h → + is-coherently-invertible g + is-coherently-invertible-right-map-triangle F H = + is-coherently-invertible-htpy + ( ( g ·l inv-htpy (is-section-map-inv-is-coherently-invertible H)) ∙h + ( inv-htpy T ·r map-inv-is-coherently-invertible H)) + ( is-coherently-invertible-comp + ( f) + ( map-inv-is-coherently-invertible H) + ( F) + ( is-coherently-invertible-map-inv-is-coherently-invertible H)) +``` + +#### The top map in a commuting triangle is coherently invertible if the other two maps are + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + (f : A → X) (g : B → X) (h : A → B) (T : f ~ g ∘ h) + where + + is-coherently-invertible-top-map-triangle : + is-coherently-invertible g → + is-coherently-invertible f → + is-coherently-invertible h + is-coherently-invertible-top-map-triangle G F = + is-coherently-invertible-htpy + ( ( inv-htpy (is-retraction-map-inv-is-coherently-invertible G) ·r h) ∙h + ( map-inv-is-coherently-invertible G ·l inv-htpy T)) + ( is-coherently-invertible-comp + ( map-inv-is-coherently-invertible G) + ( f) + ( is-coherently-invertible-map-inv-is-coherently-invertible G) + ( F)) +``` + +#### If a composite and its right factor are coherently invertible, then so is its left factor + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + where + + is-coherently-invertible-left-factor : + (g : B → X) (h : A → B) → + is-coherently-invertible (g ∘ h) → + is-coherently-invertible h → + is-coherently-invertible g + is-coherently-invertible-left-factor g h GH H = + is-coherently-invertible-right-map-triangle (g ∘ h) g h refl-htpy GH H +``` + +#### If a composite and its left factor are coherently invertible, then so is its right factor + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + where + + is-coherently-invertible-right-factor : + (g : B → X) (h : A → B) → + is-coherently-invertible g → + is-coherently-invertible (g ∘ h) → + is-coherently-invertible h + is-coherently-invertible-right-factor g h G GH = + is-coherently-invertible-top-map-triangle (g ∘ h) g h refl-htpy G GH +``` + +### Any section of a coherently invertible map is coherently invertible + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-coherently-invertible-is-section : + {f : A → B} {g : B → A} → + is-coherently-invertible f → f ∘ g ~ id → is-coherently-invertible g + is-coherently-invertible-is-section {f = f} {g} F H = + is-coherently-invertible-top-map-triangle id f g + ( inv-htpy H) + ( F) + ( is-coherently-invertible-id) +``` + +### Any retraction of a coherently invertible map is coherently invertible + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-coherently-invertible-is-retraction : + {f : A → B} {g : B → A} → + is-coherently-invertible f → (g ∘ f) ~ id → is-coherently-invertible g + is-coherently-invertible-is-retraction {f = f} {g} F H = + is-coherently-invertible-right-map-triangle id g f + ( inv-htpy H) + ( is-coherently-invertible-id) + ( F) +``` + +### If a section of `f` is coherently invertible, then `f` is coherently invertible + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) + where + + is-coherently-invertible-is-coherently-invertible-section : + (s : section f) → + is-coherently-invertible (map-section f s) → is-coherently-invertible f + is-coherently-invertible-is-coherently-invertible-section (g , G) S = + is-coherently-invertible-is-retraction S G +``` + +### If a retraction of `f` is coherently invertible, then `f` is coherently invertible + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) + where abstract - is-coherently-invertible-is-invertible : - (H : is-invertible f) → is-coherently-invertible f - pr1 (is-coherently-invertible-is-invertible H) = - map-inv-is-invertible H - pr1 (pr2 (is-coherently-invertible-is-invertible H)) = - is-section-map-inv-is-invertible H - pr1 (pr2 (pr2 (is-coherently-invertible-is-invertible H))) = - is-retraction-map-inv-is-invertible H - pr2 (pr2 (pr2 (is-coherently-invertible-is-invertible H))) = - coherence-map-inv-is-invertible H + is-coherently-invertible-is-coherently-invertible-retraction : + (r : retraction f) → + is-coherently-invertible (map-retraction f r) → is-coherently-invertible f + is-coherently-invertible-is-coherently-invertible-retraction (g , G) R = + is-coherently-invertible-is-section R G +``` + +### Any section of a coherently invertible map is homotopic to its inverse + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : coherently-invertible-map A B) + where + + htpy-map-inv-coherently-invertible-map-section : + (f : section (map-coherently-invertible-map e)) → + map-inv-coherently-invertible-map e ~ + map-section (map-coherently-invertible-map e) f + htpy-map-inv-coherently-invertible-map-section = + htpy-map-inv-invertible-map-section + ( invertible-map-coherently-invertible-map e) ``` +### Any retraction of a coherently invertible map is homotopic to its inverse + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : coherently-invertible-map A B) + where + + htpy-map-inv-coherently-invertible-map-retraction : + (f : retraction (map-coherently-invertible-map e)) → + map-inv-coherently-invertible-map e ~ + map-retraction (map-coherently-invertible-map e) f + htpy-map-inv-coherently-invertible-map-retraction = + htpy-map-inv-invertible-map-retraction + ( invertible-map-coherently-invertible-map e) +``` + +## References + +1. Egbert Rijke, _Introduction to Homotopy Type Theory_ (2022) + ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082)) +2. Univalent Foundations Project, _Homotopy Type Theory – Univalent Foundations + of Mathematics_ (2013) ([website](https://homotopytypetheory.org/book/), + [arXiv:1308.0729](https://arxiv.org/abs/1308.0729)) + ## See also - For the notion of biinvertible maps see @@ -179,3 +1401,9 @@ module _ [`foundation.contractible-maps`](foundation.contractible-maps.md). - For the notion of path-split maps see [`foundation.path-split-maps`](foundation.path-split-maps.md). + +## External links + +- [Adjoint equivalences](https://1lab.dev/1Lab.Equiv.HalfAdjoint.html) at 1lab +- [adjoint equivalence](https://ncatlab.org/nlab/show/adjoint+equivalence) at + $n$Lab diff --git a/src/foundation-core/commuting-squares-of-homotopies.lagda.md b/src/foundation-core/commuting-squares-of-homotopies.lagda.md index 1af11c0a1d..7c3e1c6877 100644 --- a/src/foundation-core/commuting-squares-of-homotopies.lagda.md +++ b/src/foundation-core/commuting-squares-of-homotopies.lagda.md @@ -8,9 +8,11 @@ module foundation-core.commuting-squares-of-homotopies where ```agda open import foundation.universe-levels +open import foundation.whiskering-homotopies-composition -open import foundation-core.equivalences +open import foundation-core.commuting-squares-of-identifications open import foundation-core.homotopies +open import foundation-core.whiskering-homotopies-concatenation ``` @@ -51,3 +53,1638 @@ module _ coherence-square-homotopies' = top ∙h right ~ left ∙h bottom ``` + +### Horizontally constant squares + +{{#concept "Horizontally constant squares" Disambiguation="homotopies" Agda=horizontal-refl-coherence-square-homotopies}} +are commuting squares of homotopies of the form + +```text + refl-htpy + f ----------> f + | | + H | | H + ∨ ∨ + g ----------> g. + refl-htpy +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g : (x : A) → B x} (H : f ~ g) + where + + horizontal-refl-coherence-square-homotopies : + coherence-square-homotopies refl-htpy H H refl-htpy + horizontal-refl-coherence-square-homotopies x = + horizontal-refl-coherence-square-identifications (H x) +``` + +### Vertically constant squares + +{{#concept "Vertically constant squares" Disambiguation="homotopies" Agda=vertical-refl-coherence-square-homotopies}} +are commuting squares of homotopies of the form + +```text + H + f -----> g + | | + refl-htpy | | refl-htpy + ∨ ∨ + f -----> g. + H +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g : (x : A) → B x} (H : f ~ g) + where + + vertical-refl-coherence-square-homotopies : + coherence-square-homotopies H refl-htpy refl-htpy H + vertical-refl-coherence-square-homotopies x = + vertical-refl-coherence-square-identifications (H x) +``` + +## Operations + +### Inverting squares of homotopies horizontally + +Given a commuting square of homotopies + +```text + top + f -------> g + | | + left | | right + ∨ ∨ + h -------> i, + bottom +``` + +the square of homotopies + +```text + top⁻¹ + g ------------> f + | | + right | | left + ∨ ∨ + i ------------> h + bottom⁻¹ +``` + +commutes. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + where + + horizontal-inv-coherence-square-homotopies : + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + coherence-square-homotopies top left right bottom → + coherence-square-homotopies (inv-htpy top) right left (inv-htpy bottom) + horizontal-inv-coherence-square-homotopies top left right bottom H x = + horizontal-inv-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) +``` + +### Inverting squares of homotopies vertically + +Given a commuting square of homotopies + +```text + top + f -------> g + | | + left | | right + ∨ ∨ + h -------> i, + bottom +``` + +the square of homotopies + +```text + bottom + h -------> i + | | + left⁻¹ | | right⁻¹ + ∨ ∨ + f -------> g + top +``` + +commutes. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + where + + vertical-inv-coherence-square-homotopies : + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + coherence-square-homotopies top left right bottom → + coherence-square-homotopies bottom (inv-htpy left) (inv-htpy right) top + vertical-inv-coherence-square-homotopies top left right bottom H x = + vertical-inv-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) +``` + +### Functions acting on squares of homotopies + +Given a commuting square of homotopies + +```text + top + f -------> g + | | + left | | right + ∨ ∨ + h -------> i + bottom +``` + +in `(x : A) → B x`, and given a dependent map `F : {x : A} → B x → C x`, the +square of homotopies + +```text + F ·l top + f f -----------> f g + | | + F ·l left | | F ·l right + ∨ ∨ + h -------------> i + F ·l bottom +``` + +commutes. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + {f g h i : (x : A) → B x} + (F : {x : A} → B x → C x) + where + + map-coherence-square-homotopies : + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + coherence-square-homotopies top left right bottom → + coherence-square-homotopies + ( F ·l top) + ( F ·l left) + ( F ·l right) + ( F ·l bottom) + map-coherence-square-homotopies top left right bottom H x = + map-coherence-square-identifications + ( F) + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) +``` + +Similarly we may whisker it on the right. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : B → UU l3} + {f g h i : (y : B) → C y} + where + + right-whisker-comp-coherence-square-homotopies : + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + (F : A → B) → + coherence-square-homotopies top left right bottom → + coherence-square-homotopies + ( top ·r F) + ( left ·r F) + ( right ·r F) + ( bottom ·r F) + right-whisker-comp-coherence-square-homotopies top left right bottom F α = + α ·r F +``` + +### Concatenating homotopies of edges and coherences of commuting squares of homotopies + +Consider a commuting square of homotopies and a homotopy of one of the four +sides with another homotopy, as for example in the diagram below: + +```text + top + a ---------> b + | | | + left | right |~| right' + ∨ ∨ ∨ + c ---------> d. + bottom +``` + +Then any homotopy witnessing that the square commutes can be concatenated with +the homotopy on the side, to obtain a new commuting square of homotopies. + +**Note.** To avoid cyclic module dependencies we will give direct proofs that +concatenating homotopies of edges of a square with the coherence of its +commutativity is an equivalence. + +#### Concatenating homotopies of the top edge with a coherence of a commuting square of homotopies + +Consider a commuting diagram of homotopies + +```text + top' + -------> + f -------> g + | top | + left | | right + ∨ ∨ + h -------> i. + bottom +``` + +with a homotopy `top ~ top'`. Then we get maps back and forth + +```text + top top' + f -------> g f -------> g + | | | | + left | | right ↔ left | | right + ∨ ∨ ∨ ∨ + h -------> i h -------> i. + bottom bottom +``` + +We record that this construction is an equivalence in +[`foundation.commuting-squares-of-homotopies`](foundation.commuting-squares-of-homotopies.md). + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + {top' : f ~ g} (s : top ~ top') + where + + concat-top-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom → + coherence-square-homotopies top' left right bottom + concat-top-homotopy-coherence-square-homotopies H x = + concat-top-identification-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( s x) + ( H x) + + inv-concat-top-homotopy-coherence-square-homotopies : + coherence-square-homotopies top' left right bottom → + coherence-square-homotopies top left right bottom + inv-concat-top-homotopy-coherence-square-homotopies H x = + inv-concat-top-identification-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( s x) + ( H x) +``` + +#### Concatenating homotopies of the left edge with a coherence of a commuting square of homotopies + +Consider a commuting diagram of homotopies + +```text + top + f -------> g + | | | + left' | | left | right + ∨ ∨ ∨ + h -------> i. + bottom +``` + +with a homotopy `left ~ left'`. Then we get maps back and forth + +```text + top top + f -------> g f -------> g + | | | | + left | | right ↔ left' | | right + ∨ ∨ ∨ ∨ + h -------> i h -------> i. + bottom bottom +``` + +We record that this construction is an equivalence in +[`foundation.commuting-squares-of-homotopies`](foundation.commuting-squares-of-homotopies.md). + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + {left' : f ~ h} (s : left ~ left') + where + + concat-left-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom → + coherence-square-homotopies top left' right bottom + concat-left-homotopy-coherence-square-homotopies H x = + concat-left-identification-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( s x) + ( H x) + + inv-concat-left-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left' right bottom → + coherence-square-homotopies top left right bottom + inv-concat-left-homotopy-coherence-square-homotopies H x = + inv-concat-left-identification-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( s x) + ( H x) +``` + +#### Concatenating homotopies of the right edge with a coherence of a commuting square of homotopies + +Consider a commuting diagram of homotopies + +```text + top + f -------> g + | | | + left | right | | right' + ∨ ∨ ∨ + h -------> i. + bottom +``` + +with a homotopy `right ~ right'`. Then we get maps back and forth + +```text + top top + f -------> g f -------> g + | | | | + left | | right ↔ left | | right' + ∨ ∨ ∨ ∨ + h -------> i h -------> i. + bottom bottom +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + {right' : g ~ i} (s : right ~ right') + where + + concat-right-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom → + coherence-square-homotopies top left right' bottom + concat-right-homotopy-coherence-square-homotopies H x = + concat-right-identification-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( s x) + ( H x) + + inv-concat-right-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right' bottom → + coherence-square-homotopies top left right bottom + inv-concat-right-homotopy-coherence-square-homotopies H x = + inv-concat-right-identification-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( s x) + ( H x) +``` + +We record that this construction is an equivalence in +[`foundation.commuting-squares-of-homotopies`](foundation.commuting-squares-of-homotopies.md). + +#### Concatenating homotopies of the bottom edge with a coherence of a commuting square of homotopies + +Consider a commuting diagram of homotopies + +```text + top + f -------> g + | | + left | | right + ∨ bottom ∨ + h -------> i. + -------> + bottom' +``` + +with a homotopy `bottom ~ bottom'`. Then we get maps back and forth + +```text + top top + f -------> g f -------> g + | | | | + left | | right ↔ left | | right + ∨ ∨ ∨ ∨ + h -------> i h -------> i. + bottom bottom' +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + {bottom' : h ~ i} (s : bottom ~ bottom') + where + + concat-bottom-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom → + coherence-square-homotopies top left right bottom' + concat-bottom-homotopy-coherence-square-homotopies H x = + concat-bottom-identification-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( s x) + ( H x) + + inv-concat-bottom-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom' → + coherence-square-homotopies top left right bottom + inv-concat-bottom-homotopy-coherence-square-homotopies H x = + inv-concat-bottom-identification-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( s x) + ( H x) +``` + +We record that this construction is an equivalence in +[`foundation.commuting-squares-of-homotopies`](foundation.commuting-squares-of-homotopies.md). + +### Whiskering and splicing coherences of commuting squares of homotopies with respect to concatenation of homotopies + +Given a commuting square of homotopies + +```text + top + f -------> g + | | + left | | right + ∨ ∨ + h -------> i, + bottom +``` + +we may consider four ways of attaching new homotopies to it: + +1. Prepending `H : u ~ f` to the left gives us a commuting square + + ```text + H ∙h top + u -------> g + | | + H ∙h left | | right + ∨ ∨ + h -------> i. + bottom + ``` + + More precisely, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ ((H ∙h left) ∙h bottom ~ (H ∙h top) ∙h right). + ``` + +2. Appending a homotopy `H : i ~ u` to the right gives a commuting square of + homotopies + + ```text + top + f ------------> g + | | + left | | right ∙h H + ∨ ∨ + h ------------> u. + bottom ∙h H + ``` + + More precisely, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ (left ∙h (bottom ∙h H) ~ top ∙h (right ∙h H)). + ``` + +3. Splicing a homotopy `H : h ~ u` and its inverse into the middle gives a + commuting square of homotopies + + ```text + top + f --------------> g + | | + left ∙h H | | right + ∨ ∨ + u --------------> i. + H⁻¹ ∙h bottom + ``` + + More precisely, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ ((left ∙h H) ∙h (H⁻¹ ∙h bottom) ~ top ∙h right). + ``` + + Similarly, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ ((left ∙h H⁻¹) ∙h (H ∙h bottom) ~ top ∙h right). + ``` + +4. Splicing a homotopy `H : g ~ u` and its inverse into the middle gives a + commuting square of homotopies + + ```text + top ∙h H + f --------> u + | | + left | | H⁻¹ ∙h right + ∨ ∨ + h --------> i. + bottom + ``` + + More precisely, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ (left ∙h bottom ~ (top ∙h H) ∙h (H⁻¹ ∙h right)). + ``` + + Similarly, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ (left ∙h bottom ~ (top ∙h H⁻¹) ∙h (H ∙h right)). + ``` + +These operations are useful in proofs involving homotopy algebra, because taking +`equiv-right-whisker-concat-coherence-square-homotopies` as an example, it +provides us with two maps: the forward direction states +`(H ∙h r ~ K ∙h s) → (H ∙h (r ∙h t)) ~ K ∙h (s ∙h t))`, which allows one to +append a homotopy without needing to reassociate on the right, and the backwards +direction conversely allows one to cancel out a homotopy in parentheses. + +#### Left whiskering coherences of commuting squares of homotopies + +For any homotopy `H : u ~ f` we obtain maps back and forth + +```text + top H ∙h top + f -------> g u -------> g + | | | | + left | | right ↔ H ∙h left | | right + ∨ ∨ ∨ ∨ + h -------> i h -------> i + bottom bottom +``` + +of coherences of commuting squares of homotopies. We show in +[`foundation.commuting-squares-of-homotopies`](foundation.commuting-squares-of-homotopies.md) +that these maps are equivalences. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + where + + left-whisker-concat-coherence-square-homotopies : + {u : (x : A) → B x} (H : u ~ f) + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + coherence-square-homotopies top left right bottom → + coherence-square-homotopies (H ∙h top) (H ∙h left) right bottom + left-whisker-concat-coherence-square-homotopies + H top left right bottom coh x = + left-whisker-concat-coherence-square-identifications + ( H x) + ( top x) + ( left x) + ( right x) + ( bottom x) + ( coh x) + + left-unwhisker-concat-coherence-square-homotopies : + {u : (x : A) → B x} (H : u ~ f) + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + coherence-square-homotopies (H ∙h top) (H ∙h left) right bottom → + coherence-square-homotopies top left right bottom + left-unwhisker-concat-coherence-square-homotopies + H top left right bottom coh x = + left-unwhisker-concat-coherence-square-identifications + ( H x) + ( top x) + ( left x) + ( right x) + ( bottom x) + ( coh x) +``` + +#### Right whiskering coherences of commuting squares of homotopies + +For any homotopy `H : i ~ u` we obtain maps back and forth + +```text + top top + f -------> g f ------------> g + | | | | + left | | right ↔ left | | right ∙h H + ∨ ∨ ∨ ∨ + h -------> i h ------------> i + bottom bottom ∙h H +``` + +of coherences of commuting squares of homotopies. We show in +[`foundation.commuting-squares-of-homotopies`](foundation.commuting-squares-of-homotopies.md) +that these maps are equivalences. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + where + + right-whisker-concat-coherence-square-homotopies : + coherence-square-homotopies top left right bottom → + {u : (x : A) → B x} (H : i ~ u) → + coherence-square-homotopies top left (right ∙h H) (bottom ∙h H) + right-whisker-concat-coherence-square-homotopies coh H x = + right-whisker-concat-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( coh x) + ( H x) + + right-unwhisker-cohernece-square-homotopies : + {u : (x : A) → B x} (H : i ~ u) → + coherence-square-homotopies top left (right ∙h H) (bottom ∙h H) → + coherence-square-homotopies top left right bottom + right-unwhisker-cohernece-square-homotopies H coh x = + right-unwhisker-cohernece-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) + ( coh x) +``` + +### Double whiskering of commuting squares of homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h u v i : (x : A) → B x} + where + + double-whisker-coherence-square-homotopies : + (p : f ~ g) + (top : g ~ u) (left : g ~ h) (right : u ~ v) (bottom : h ~ v) + (s : v ~ i) → + coherence-square-homotopies top left right bottom → + coherence-square-homotopies + ( p ∙h top) + ( p ∙h left) + ( right ∙h s) + ( bottom ∙h s) + double-whisker-coherence-square-homotopies p top left right bottom q H = + left-whisker-concat-coherence-square-homotopies p top left + ( right ∙h q) + ( bottom ∙h q) + ( right-whisker-concat-coherence-square-homotopies + ( top) + ( left) + ( right) + ( bottom) + ( H) + ( q)) +``` + +#### Left splicing coherences of commuting squares of homotopies + +For any inverse pair of homotopies `H : g ~ u` and `K : u ~ g` equipped with +`α : inv-htpy H ~ K` we obtain maps back and forth + +```text + top top + f -------> g f -----------> g + | | | | + left | | right ↔ left ∙h H | | right + ∨ ∨ ∨ ∨ + h -------> i u -----------> i + bottom K ∙h bottom +``` + +of coherences of commuting squares of homotopies. We show in +[`foundation.commuting-squares-of-homotopies`](foundation.commuting-squares-of-homotopies.md) +that these maps are equivalences. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + where + + left-splice-coherence-square-homotopies : + {u : (x : A) → B x} (H : h ~ u) (K : u ~ h) (α : inv-htpy H ~ K) → + coherence-square-homotopies top left right bottom → + coherence-square-homotopies top (left ∙h H) right (K ∙h bottom) + left-splice-coherence-square-homotopies H K α coh x = + left-splice-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) + ( K x) + ( α x) + ( coh x) + + left-unsplice-coherence-square-homotopies : + {u : (x : A) → B x} (H : h ~ u) (K : u ~ h) (α : inv-htpy H ~ K) → + coherence-square-homotopies top (left ∙h H) right (K ∙h bottom) → + coherence-square-homotopies top left right bottom + left-unsplice-coherence-square-homotopies H K α coh x = + left-unsplice-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) + ( K x) + ( α x) + ( coh x) +``` + +#### Right splicing coherences of commuting squares of homotopies + +For any inverse pair of homotopies `H : g ~ u` and `K : u ~ g` equipped with +`α : inv-htpy H ~ K` we obtain maps back and forth + +```text + top top ∙h H + f -------> g f --------> u + | | | | + left | | right ↔ left | | K ∙h right + ∨ ∨ ∨ ∨ + h -------> i h --------> i + bottom bottom +``` + +of coherences of commuting squares of homotopies. We show in +[`foundation.commuting-squares-of-homotopies`](foundation.commuting-squares-of-homotopies.md) +that these maps are equivalences. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + where + + right-splice-coherence-square-homotopies : + {u : (x : A) → B x} (H : g ~ u) (K : u ~ g) (α : inv-htpy H ~ K) → + coherence-square-homotopies top left right bottom → + coherence-square-homotopies (top ∙h H) left (inv-htpy H ∙h right) bottom + right-splice-coherence-square-homotopies H K α coh x = + right-splice-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) + ( K x) + ( α x) + ( coh x) + + right-unsplice-coherence-square-homotopies : + {u : (x : A) → B x} (H : g ~ u) (K : u ~ g) (α : inv-htpy H ~ K) → + coherence-square-homotopies (top ∙h H) left (inv-htpy H ∙h right) bottom → + coherence-square-homotopies top left right bottom + right-unsplice-coherence-square-homotopies H K α coh x = + right-unsplice-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) + ( K x) + ( α x) + ( coh x) +``` + +### Horizontally pasting squares of homotopies + +Consider two squares of homotopies as in the diagram + +```text + top-left top-right + a -------------> b -------------> c + | | | + left | | middle | right + ∨ ∨ ∨ + d -------------> e -------------> f + bottom-left bottom-right +``` + +with `H : left ∙h bottom-left ~ top-left ∙h middle` and +`K : middle ∙h bottom-right ~ top-right ∙h right`. Then the outer square +commutes. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a b c d e f : (x : A) → B x} + (top-left : a ~ b) (top-right : b ~ c) + (left : a ~ d) (middle : b ~ e) (right : c ~ f) + (bottom-left : d ~ e) (bottom-right : e ~ f) + where + + horizontal-pasting-coherence-square-homotopies : + coherence-square-homotopies top-left left middle bottom-left → + coherence-square-homotopies top-right middle right bottom-right → + coherence-square-homotopies + (top-left ∙h top-right) left right (bottom-left ∙h bottom-right) + horizontal-pasting-coherence-square-homotopies H K x = + horizontal-pasting-coherence-square-identifications + ( top-left x) + ( top-right x) + ( left x) + ( middle x) + ( right x) + ( bottom-left x) + ( bottom-right x) + ( H x) + ( K x) +``` + +### Vertically pasting squares of homotopies + +Consider two squares of homotopies as in the diagram + +```text + top + a --------> b + | | + top-left | | top-right + ∨ middle ∨ + c --------> d + | | + bottom-left | | bottom-right + ∨ ∨ + e --------> f + bottom +``` + +with `H : top-left ∙h middle ~ top ∙h top-right` and +`K : bottom-left ∙h bottom ~ middle ∙h bottom-right`. Then the outer square +commutes. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a b c d e f : (x : A) → B x} + (top : a ~ b) (top-left : a ~ c) (top-right : b ~ d) + (middle : c ~ d) (bottom-left : c ~ e) (bottom-right : d ~ f) + (bottom : e ~ f) + where + + vertical-pasting-coherence-square-homotopies : + coherence-square-homotopies top top-left top-right middle → + coherence-square-homotopies middle bottom-left bottom-right bottom → + coherence-square-homotopies + top (top-left ∙h bottom-left) (top-right ∙h bottom-right) bottom + vertical-pasting-coherence-square-homotopies H K x = + vertical-pasting-coherence-square-identifications + ( top x) + ( top-left x) + ( top-right x) + ( middle x) + ( bottom-left x) + ( bottom-right x) + ( bottom x) + ( H x) + ( K x) +``` + +## Properties + +### Left unit law for horizontal pasting of commuting squares of homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + where + + left-unit-law-horizontal-pasting-coherence-square-homotopies : + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + (H : coherence-square-homotopies top left right bottom) → + horizontal-pasting-coherence-square-homotopies + ( refl-htpy) + ( top) + ( left) + ( left) + ( right) + ( refl-htpy) + ( bottom) + ( horizontal-refl-coherence-square-homotopies left) + ( H) ~ + H + left-unit-law-horizontal-pasting-coherence-square-homotopies + top left right bottom H x = + left-unit-law-horizontal-pasting-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) +``` + +### Right unit law for horizontal pasting of commuting squares of homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + where + + right-unit-law-horizontal-pasting-coherence-square-homotopies : + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + (H : coherence-square-homotopies top left right bottom) → + horizontal-pasting-coherence-square-homotopies + ( top) + ( refl-htpy) + ( left) + ( right) + ( right) + ( bottom) + ( refl-htpy) + ( H) + ( horizontal-refl-coherence-square-homotopies right) ∙h + right-whisker-concat-htpy right-unit-htpy right ~ + left-whisker-concat-htpy left right-unit-htpy ∙h H + right-unit-law-horizontal-pasting-coherence-square-homotopies + top left right bottom H x = + right-unit-law-horizontal-pasting-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) +``` + +### Left unit law for vertical pasting of commuting squares of homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + where + + left-unit-law-vertical-pasting-coherence-square-homotopies : + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + (H : coherence-square-homotopies top left right bottom) → + vertical-pasting-coherence-square-homotopies + ( top) + ( refl-htpy) + ( refl-htpy) + ( top) + ( left) + ( right) + ( bottom) + ( vertical-refl-coherence-square-homotopies top) + ( H) ~ + H + left-unit-law-vertical-pasting-coherence-square-homotopies + top left right bottom H x = + left-unit-law-vertical-pasting-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) +``` + +### Right unit law for vertical pasting of commuting squares of homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + where + + right-unit-law-vertical-pasting-coherence-square-homotopies : + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + (H : coherence-square-homotopies top left right bottom) → + vertical-pasting-coherence-square-homotopies + ( top) + ( left) + ( right) + ( bottom) + ( refl-htpy) + ( refl-htpy) + ( bottom) + ( H) + ( vertical-refl-coherence-square-homotopies bottom) ∙h + left-whisker-concat-htpy top right-unit-htpy ~ + right-whisker-concat-htpy right-unit-htpy bottom ∙h H + right-unit-law-vertical-pasting-coherence-square-homotopies + top left right bottom H x = + right-unit-law-vertical-pasting-coherence-square-identifications + ( top x) + ( left x) + ( right x) + ( bottom x) + ( H x) +``` + +### Computing the right whiskering of a vertically constant square with a homotopy + +Consider the vertically constant square of homotopies + +```text + H + f -----> g + | | + refl | | refl + ∨ ∨ + f -----> g + H +``` + +at a homotopy `H : f ~ g`, and consider a homotopy `K : g ~ h`. Then the right +whiskering of the above square with `K` is the commuting square of homotopies + +```text + H + f -------> g + | | + refl | refl | K + ∨ ∨ + f -------> h + H ∙h K +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h : (x : A) → B x} + where + + right-whisker-concat-vertical-refl-coherence-square-homotopies : + (H : f ~ g) (K : g ~ h) → + right-whisker-concat-coherence-square-homotopies H refl-htpy refl-htpy H + ( vertical-refl-coherence-square-homotopies H) + ( K) ~ + refl-htpy + right-whisker-concat-vertical-refl-coherence-square-homotopies H K x = + right-whisker-concat-vertical-refl-coherence-square-identifications + ( H x) + ( K x) +``` + +### Computing the right whiskering of a horizontally constant square with a homotopy + +Consider a horizontally constant commuting square of homotopies + +```text + refl-htpy + f ----------> f + | | + H | | H + ∨ ∨ + g ----------> g + refl-htpy +``` + +at a homotopy `H` and consider a homotopy `K : g ~ h`. Then the right whiskering +of the above square with `K` is the square + +```text + refl-htpy + f ----------> f + | | + H | refl-htpy | H ∙h K + ∨ ∨ + g ----------> h. + K +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h : (x : A) → B x} + where + + right-whisker-concat-horizontal-refl-coherence-square-homotopies : + (H : f ~ g) (K : g ~ h) → + right-whisker-concat-coherence-square-homotopies refl-htpy H H refl-htpy + ( horizontal-refl-coherence-square-homotopies H) + ( K) ~ + refl-htpy + right-whisker-concat-horizontal-refl-coherence-square-homotopies H K x = + right-whisker-concat-horizontal-refl-coherence-square-identifications + ( H x) + ( K x) +``` + +### Computing the left whiskering of a horizontally constant square with a homotopy + +Consider a homotopy `H : f ~ g` and a horizontally constant commuting square of +homotopies + +```text + refl-htpy + g ----------> g + | | + K | | K + ∨ ∨ + h ----------> h + refl-htpy +``` + +at a homotopy `K : g ~ h`. The the left whiskering of the above square with `H` +is the commuting square + +```text + K ∙h refl-htpy + f -----------------------------------------------------------------> g + | | + K ∙h H | right-unit-htpy ∙h (right-whisker-concat-htpy right-unit-htpy H)⁻¹ | H + ∨ ∨ + h -----------------------------------------------------------------> h. + refl-htpy +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h : (x : A) → B x} + where + + left-whisker-concat-horizontal-refl-coherence-square-homotopies : + (H : f ~ g) (K : g ~ h) → + left-whisker-concat-coherence-square-homotopies H refl-htpy K K refl-htpy + ( horizontal-refl-coherence-square-homotopies K) ∙h + right-whisker-concat-htpy right-unit-htpy K ~ + right-unit-htpy + left-whisker-concat-horizontal-refl-coherence-square-homotopies H K x = + left-whisker-concat-horizontal-refl-coherence-square-identifications + ( H x) + ( K x) + + left-whisker-concat-horizontal-refl-coherence-square-homotopies' : + (H : f ~ g) (K : g ~ h) → + left-whisker-concat-coherence-square-homotopies H refl-htpy K K refl-htpy + ( horizontal-refl-coherence-square-homotopies K) ~ + right-unit-htpy ∙h inv-htpy (right-whisker-concat-htpy right-unit-htpy K) + left-whisker-concat-horizontal-refl-coherence-square-homotopies' H K x = + left-whisker-concat-horizontal-refl-coherence-square-identifications' + ( H x) + ( K x) +``` + +### Computing the left whiskering of a vertically constant square with a homotopy + +Consider the vertically constant square of homotopies + +```text + K + g -----> h + | | + refl-htpy | | refl-htpy + ∨ ∨ + g -----> h + K +``` + +at a homotopy `K : g ~ h` and consider a homotopy `H : f ~ g`. Then the left +whiskering of the above square with `H` is the square + +```text + H ∙h K + f ----------------------------------------------------------> h + | | + H ∙h refl-htpy | right-whisker-concat-htpy right-unit-htpy K ∙h right-unit⁻¹ | refl-htpy + ∨ ∨ + g ----------------------------------------------------------> h. + K +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h : (x : A) → B x} + where + + left-whisker-concat-vertical-refl-coherence-square-homotopies : + (H : f ~ g) (K : g ~ h) → + left-whisker-concat-coherence-square-homotopies H K refl-htpy refl-htpy K + ( vertical-refl-coherence-square-homotopies K) ∙h + right-unit-htpy ~ + right-whisker-concat-htpy right-unit-htpy K + left-whisker-concat-vertical-refl-coherence-square-homotopies H K x = + left-whisker-concat-vertical-refl-coherence-square-identifications + ( H x) + ( K x) + + left-whisker-concat-vertical-refl-coherence-square-homotopies' : + (H : f ~ g) (K : g ~ h) → + left-whisker-concat-coherence-square-homotopies H K refl-htpy refl-htpy K + ( vertical-refl-coherence-square-homotopies K) ~ + right-whisker-concat-htpy right-unit-htpy K ∙h inv-htpy right-unit-htpy + left-whisker-concat-vertical-refl-coherence-square-homotopies' H K x = + left-whisker-concat-vertical-refl-coherence-square-identifications' + ( H x) + ( K x) +``` + +### Left whiskering horizontal concatenations of squares with homotopies + +Consider a commuting diagram of homotopies of the form + +```text + top-left top-right + a -------------> c -------------> e + | | | + left | | middle | right + ∨ ∨ ∨ + b -------------> d -------------> f + bottom-left bottom-right +``` + +and consider a homotopy `H : f ~ a`. Then the left whiskering of `H` and the +horizontal concatenation of coherences of commuting squares is up to +associativity the horizontal concatenation of the squares + +```text + H ∙h top-left top-right + u -------------> c -------------> e + | | | + H ∙h left | | middle | right + ∨ ∨ ∨ + b -------------> d -------------> f + bottom-left bottom-right +``` + +where the left square is the left whiskering of `H` and the original left +square. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + left-whisker-concat-horizontal-pasting-coherence-square-homotopies : + {u a b c d e f : (x : A) → B x} (H : u ~ a) + (top-left : a ~ c) (top-right : c ~ e) + (left : a ~ b) (middle : c ~ d) (right : e ~ f) + (bottom-left : b ~ d) (bottom-right : d ~ f) + (l : coherence-square-homotopies top-left left middle bottom-left) + (r : coherence-square-homotopies top-right middle right bottom-right) → + left-whisker-concat-coherence-square-homotopies H + ( top-left ∙h top-right) + ( left) + ( right) + ( bottom-left ∙h bottom-right) + ( horizontal-pasting-coherence-square-homotopies + ( top-left) + ( top-right) + ( left) + ( middle) + ( right) + ( bottom-left) + ( bottom-right) + ( l) + ( r)) ~ + horizontal-pasting-coherence-square-homotopies + ( H ∙h top-left) + ( top-right) + ( H ∙h left) + ( middle) + ( right) + ( bottom-left) + ( bottom-right) + ( left-whisker-concat-coherence-square-homotopies H + ( top-left) + ( left) + ( middle) + ( bottom-left) + ( l)) + ( r) ∙h + right-whisker-concat-htpy + ( assoc-htpy H top-left top-right) + ( right) + left-whisker-concat-horizontal-pasting-coherence-square-homotopies + H top-left top-right left middle right bottom-left bottom-right l r x = + left-whisker-concat-horizontal-pasting-coherence-square-identifications + ( H x) + ( top-left x) + ( top-right x) + ( left x) + ( middle x) + ( right x) + ( bottom-left x) + ( bottom-right x) + ( l x) + ( r x) +``` + +### Left whiskering vertical concatenations of squares with homotopies + +Consider two squares of homotopies as in the diagram + +```text + top + a --------> b + | | + top-left | | top-right + ∨ middle ∨ + c --------> d + | | + bottom-left | | bottom-right + ∨ ∨ + e --------> f + bottom +``` + +and consider a homotopy `H : f ~ a`. Then the left whiskering of `H` with the +vertical pasting of the two squares above is up to associativity the vertical +pasting of the squares + +```text + H ∙h top + u --------> b + | | + H ∙h top-left | | top-right + ∨ middle ∨ + c --------> d + | | + bottom-left | | bottom-right + ∨ ∨ + e --------> f. + bottom +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + left-whisker-concat-vertical-concat-coherence-square-homotopies : + {u a b c d e f : (x : A) → B x} (H : u ~ a) → + (top : a ~ b) (top-left : a ~ c) (top-right : b ~ d) (middle : c ~ d) + (bottom-left : c ~ e) (bottom-right : d ~ f) (bottom : e ~ f) + (t : coherence-square-homotopies top top-left top-right middle) → + (b : + coherence-square-homotopies middle bottom-left bottom-right bottom) → + right-whisker-concat-htpy (assoc-htpy H top-left bottom-left) bottom ∙h + left-whisker-concat-coherence-square-homotopies H + ( top) + ( top-left ∙h bottom-left) + ( top-right ∙h bottom-right) + ( bottom) + ( vertical-pasting-coherence-square-homotopies + ( top) + ( top-left) + ( top-right) + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( t) + ( b)) ~ + vertical-pasting-coherence-square-homotopies + ( H ∙h top) + ( H ∙h top-left) + ( top-right) + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( left-whisker-concat-coherence-square-homotopies H + ( top) + ( top-left) + ( top-right) + ( middle) + ( t)) + ( b) + left-whisker-concat-vertical-concat-coherence-square-homotopies + H top top-left top-right middle bottom-left bottom-right bottom t b x = + left-whisker-concat-vertical-concat-coherence-square-identifications + ( H x) + ( top x) + ( top-left x) + ( top-right x) + ( middle x) + ( bottom-left x) + ( bottom-right x) + ( bottom x) + ( t x) + ( b x) +``` + +### Right whiskering horizontal pastings of commuting squares of homotopies + +Consider a commuting diagram of homotopies of the form + +```text + top-left top-right + a -------------> c -------------> e + | | | + left | | middle | right + ∨ ∨ ∨ + b -------------> d -------------> f + bottom-left bottom-right +``` + +and consider a homotopy `K : f ~ g`. Then the right whiskering of the horizontal +pasting of the squares above is up to associativity the horizontal pasting of +the squares + +```text + top-left top-right + a -------------> c ------------------> e + | | | + left | | middle | right ∙h K + ∨ ∨ ∨ + b -------------> d ------------------> g + bottom-left bottom-right ∙h K +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + right-whisker-concat-horizontal-pasting-coherence-square-homotopies : + {a b c d e f g : (x : A) → B x} + (top-left : a ~ c) (top-right : c ~ e) + (left : a ~ b) (middle : c ~ d) (right : e ~ f) + (bottom-left : b ~ d) (bottom-right : d ~ f) + (l : coherence-square-homotopies top-left left middle bottom-left) → + (r : coherence-square-homotopies top-right middle right bottom-right) → + (K : f ~ g) → + right-whisker-concat-coherence-square-homotopies + ( top-left ∙h top-right) + ( left) + ( right) + ( bottom-left ∙h bottom-right) + ( horizontal-pasting-coherence-square-homotopies + ( top-left) + ( top-right) + ( left) + ( middle) + ( right) + ( bottom-left) + ( bottom-right) + ( l) + ( r)) + ( K) ~ + left-whisker-concat-htpy left (assoc-htpy bottom-left bottom-right K) ∙h + horizontal-pasting-coherence-square-homotopies + ( top-left) + ( top-right) + ( left) + ( middle) + ( right ∙h K) + ( bottom-left) + ( bottom-right ∙h K) + ( l) + ( right-whisker-concat-coherence-square-homotopies + ( top-right) + ( middle) + ( right) + ( bottom-right) + ( r) + ( K)) + right-whisker-concat-horizontal-pasting-coherence-square-homotopies + top-left top-right left middle right bottom-left bottom-right l r K x = + right-whisker-concat-horizontal-pasting-coherence-square-identifications + ( top-left x) + ( top-right x) + ( left x) + ( middle x) + ( right x) + ( bottom-left x) + ( bottom-right x) + ( l x) + ( r x) + ( K x) +``` + +### Right whiskering vertical concatenations of squares with homotopies + +Consider two squares of homotopies as in the diagram + +```text + top + a --------> b + | | + top-left | | top-right + ∨ middle ∨ + c --------> d + | | + bottom-left | | bottom-right + ∨ ∨ + e --------> f + bottom +``` + +and consider a homotopy `K : f ~ g`. Then the right whiskering of the vertical +pasting of the two squares above with `K` is up to associativity the vertical +pasting of the squares + +```text + top + a ------------> b + | | + top-left | | top-right + ∨ middle ∨ + c ------------> d + | | + bottom-left | | bottom-right ∙h K + ∨ ∨ + e ------------> g. + bottom ∙h K +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + right-whisker-concat-vertical-pasting-coherence-square-homotopies : + {a b c d e f g : (x : A) → B x} + (top : a ~ b) (top-left : a ~ c) (top-right : b ~ d) + (middle : c ~ d) + (bottom-left : c ~ e) (bottom-right : d ~ f) (bottom : e ~ f) + (t : coherence-square-homotopies top top-left top-right middle) → + (b : + coherence-square-homotopies middle bottom-left bottom-right bottom) → + (K : f ~ g) → + right-whisker-concat-coherence-square-homotopies + ( top) + ( top-left ∙h bottom-left) + ( top-right ∙h bottom-right) + ( bottom) + ( vertical-pasting-coherence-square-homotopies + ( top) + ( top-left) + ( top-right) + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( t) + ( b)) + ( K) ∙h + left-whisker-concat-htpy top (assoc-htpy top-right bottom-right K) ~ + vertical-pasting-coherence-square-homotopies + ( top) + ( top-left) + ( top-right) + ( middle) + ( bottom-left) + ( bottom-right ∙h K) + ( bottom ∙h K) + ( t) + ( right-whisker-concat-coherence-square-homotopies + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( b) + ( K)) + right-whisker-concat-vertical-pasting-coherence-square-homotopies + top top-left top-right middle bottom-left bottom-right bottom t b K x = + right-whisker-concat-vertical-pasting-coherence-square-identifications + ( top x) + ( top-left x) + ( top-right x) + ( middle x) + ( bottom-left x) + ( bottom-right x) + ( bottom x) + ( t x) + ( b x) + ( K x) +``` diff --git a/src/foundation-core/commuting-squares-of-identifications.lagda.md b/src/foundation-core/commuting-squares-of-identifications.lagda.md new file mode 100644 index 0000000000..a3d8f14744 --- /dev/null +++ b/src/foundation-core/commuting-squares-of-identifications.lagda.md @@ -0,0 +1,1579 @@ +# Commuting squares of identifications + +```agda +module foundation-core.commuting-squares-of-identifications where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.universe-levels + +open import foundation-core.function-types +open import foundation-core.identity-types +open import foundation-core.retractions +open import foundation-core.sections +open import foundation-core.whiskering-identifications-concatenation +``` + +
+ +## Idea + +A square of [identifications](foundation-core.identity-types.md) + +```text + top + x -------> y + | | + left | | right + ∨ ∨ + z -------> w + bottom +``` + +is said to be a +{{#concept "commuting square" Disambiguation="identifications" Agda=coherence-square-identifications}} +if there is an identification `left ∙ bottom ＝ top ∙ right`. Such an +identification is called a +{{#concept "coherence" Disambiguation="commuting square of identifications" Agda=coherence-square-identifications}} +of the square. + +## Definitions + +### Commuting squares of identifications + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) + where + + coherence-square-identifications : UU l + coherence-square-identifications = left ∙ bottom ＝ top ∙ right +``` + +### Horizontally constant squares + +{{#concept "Horizontally constant squares" Disambiguation="identifications" Agda=horizontal-refl-coherence-square-identifications}} +are commuting squares of identifications of the form + +```text + refl + a -----> a + | | + p | | p + ∨ ∨ + b -----> b. + refl +``` + +```agda +module _ + {l : Level} {A : UU l} {a b : A} (p : a ＝ b) + where + + horizontal-refl-coherence-square-identifications : + coherence-square-identifications refl p p refl + horizontal-refl-coherence-square-identifications = right-unit +``` + +### Vertically constant squares + +{{#concept "Vertically constant squares" Disambiguation="identifications" Agda=vertical-refl-coherence-square-identifications}} +are commuting squares of identifications of the form + +```text + p + a -----> b + | | + refl | | refl + ∨ ∨ + a -----> b. + p +``` + +```agda +module _ + {l : Level} {A : UU l} {a b : A} (p : a ＝ b) + where + + vertical-refl-coherence-square-identifications : + coherence-square-identifications p refl refl p + vertical-refl-coherence-square-identifications = inv right-unit +``` + +## Operations + +### Inverting squares of identifications horizontally + +Given a commuting square of identifications + +```text + top + x -------> y + | | + left | | right + ∨ ∨ + z -------> w, + bottom +``` + +the square of identifications + +```text + inv top + y ------------> x + | | + right | | left + ∨ ∨ + w ------------> z + inv bottom +``` + +commutes. + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + where + + horizontal-inv-coherence-square-identifications : + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → + coherence-square-identifications top left right bottom → + coherence-square-identifications (inv top) right left (inv bottom) + horizontal-inv-coherence-square-identifications refl left right bottom coh = + inv (right-transpose-eq-concat left bottom right coh) +``` + +### Inverting squares of identifications vertically + +Given a commuting square of identifications + +```text + top + x -------> y + | | + left | | right + ∨ ∨ + z -------> w, + bottom +``` + +the square of identifications + +```text + bottom + z -------> w + | | + inv left | | inv right + ∨ ∨ + x -------> y + top +``` + +commutes. + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + where + + vertical-inv-coherence-square-identifications : + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → + coherence-square-identifications top left right bottom → + coherence-square-identifications bottom (inv left) (inv right) top + vertical-inv-coherence-square-identifications top refl right bottom coh = + right-transpose-eq-concat top right (bottom) (inv coh) +``` + +### Functions acting on squares of identifications + +Given a commuting square of identifications + +```text + top + x -------> y + | | + left | | right + ∨ ∨ + z -------> w + bottom +``` + +in a type `A`, and given a map `f : A → B`, the square of identifications + +```text + ap f top + f x -----------> f y + | | + ap f left | | ap f right + ∨ ∨ + z -------------> w + ap f bottom +``` + +commutes. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {x y z w : A} (f : A → B) + where + + map-coherence-square-identifications : + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → + coherence-square-identifications top left right bottom → + coherence-square-identifications + ( ap f top) + ( ap f left) + ( ap f right) + ( ap f bottom) + map-coherence-square-identifications refl refl _ _ coh = ap (ap f) coh +``` + +### Concatenating identifications of edges and coherences of commuting squares of identifications + +Consider a commuting square of identifications and an identification of one of +the four sides with another identification, as for example in the diagram below: + +```text + top + a ---------> b + | | | + left | right |=| right' + ∨ ∨ ∨ + c ---------> d. + bottom +``` + +Then any identification witnessing that the square commutes can be concatenated +with the identification on the side, to obtain a new commuting square of +identifications. + +**Note.** To avoid cyclic module dependencies we will give direct proofs that +concatenating identifications of edges of a square with the coherence of its +commutativity is an equivalence. + +#### Concatenating identifications of the top edge with a coherence of a commuting square of identifications + +Consider a commuting diagram of identifications + +```text + top' + -------> + x -------> y + | top | + left | | right + ∨ ∨ + z -------> w. + bottom +``` + +with an identification `top ＝ top'`. Then we get an equivalence + +```text + top top' + x -------> y x -------> y + | | | | + left | | right ≃ left | | right + ∨ ∨ ∨ ∨ + z -------> w z -------> w. + bottom bottom +``` + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) + {top' : x ＝ y} (s : top ＝ top') + where + + concat-top-identification-coherence-square-identifications : + coherence-square-identifications top left right bottom → + coherence-square-identifications top' left right bottom + concat-top-identification-coherence-square-identifications t = + t ∙ ap (concat' _ right) s + + inv-concat-top-identification-coherence-square-identifications : + coherence-square-identifications top' left right bottom → + coherence-square-identifications top left right bottom + inv-concat-top-identification-coherence-square-identifications t = + t ∙ inv (ap (concat' _ right) s) + + is-section-inv-concat-top-identification-coherence-square-identifications : + is-section + concat-top-identification-coherence-square-identifications + inv-concat-top-identification-coherence-square-identifications + is-section-inv-concat-top-identification-coherence-square-identifications = + is-section-inv-concat' (ap (concat' _ right) s) + + is-retraction-inv-concat-top-identification-coherence-square-identifications : + is-retraction + concat-top-identification-coherence-square-identifications + inv-concat-top-identification-coherence-square-identifications + is-retraction-inv-concat-top-identification-coherence-square-identifications = + is-retraction-inv-concat' (ap (concat' _ right) s) +``` + +We record that this construction is an equivalence in +[`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md). + +#### Concatenating identifications of the left edge with a coherence of a commuting square of identifications + +Consider a commuting diagram of identifications + +```text + top + x -------> y + | | | + left' | | left | right + ∨ ∨ ∨ + z -------> w. + bottom +``` + +with an identification `left ＝ left'`. Then we get an equivalence + +```text + top top + x -------> y x -------> y + | | | | + left | | right ≃ left' | | right + ∨ ∨ ∨ ∨ + z -------> w z -------> w. + bottom bottom +``` + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) + {left' : x ＝ z} (s : left ＝ left') + where + + concat-left-identification-coherence-square-identifications : + coherence-square-identifications top left right bottom → + coherence-square-identifications top left' right bottom + concat-left-identification-coherence-square-identifications t = + inv (ap (concat' _ bottom) s) ∙ t + + inv-concat-left-identification-coherence-square-identifications : + coherence-square-identifications top left' right bottom → + coherence-square-identifications top left right bottom + inv-concat-left-identification-coherence-square-identifications t = + ap (concat' _ bottom) s ∙ t + + is-section-inv-concat-left-identification-coherence-square-identifications : + is-section + concat-left-identification-coherence-square-identifications + inv-concat-left-identification-coherence-square-identifications + is-section-inv-concat-left-identification-coherence-square-identifications = + is-retraction-inv-concat (ap (concat' _ bottom) s) + + is-retraction-inv-concat-left-identification-coherence-square-identifications : + is-retraction + concat-left-identification-coherence-square-identifications + inv-concat-left-identification-coherence-square-identifications + is-retraction-inv-concat-left-identification-coherence-square-identifications = + is-section-inv-concat (ap (concat' _ bottom) s) +``` + +We record that this construction is an equivalence in +[`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md). + +#### Concatenating identifications of the right edge with a coherence of a commuting square of identifications + +Consider a commuting diagram of identifications + +```text + top + x -------> y + | | | + left | right | | right' + ∨ ∨ ∨ + z -------> w. + bottom +``` + +with an identification `right ＝ right'`. Then we get an equivalence + +```text + top top + x -------> y x -------> y + | | | | + left | | right ≃ left | | right' + ∨ ∨ ∨ ∨ + z -------> w z -------> w. + bottom bottom +``` + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) + {right' : y ＝ w} (s : right ＝ right') + where + + concat-right-identification-coherence-square-identifications : + coherence-square-identifications top left right bottom → + coherence-square-identifications top left right' bottom + concat-right-identification-coherence-square-identifications t = + t ∙ ap (concat top _) s + + inv-concat-right-identification-coherence-square-identifications : + coherence-square-identifications top left right' bottom → + coherence-square-identifications top left right bottom + inv-concat-right-identification-coherence-square-identifications t = + t ∙ inv (ap (concat top _) s) + + is-section-inv-concat-right-identification-coherence-square-identifications : + is-section + concat-right-identification-coherence-square-identifications + inv-concat-right-identification-coherence-square-identifications + is-section-inv-concat-right-identification-coherence-square-identifications = + is-section-inv-concat' (ap (concat top _) s) + + is-retraction-inv-concat-right-identification-coherence-square-identifications : + is-retraction + concat-right-identification-coherence-square-identifications + inv-concat-right-identification-coherence-square-identifications + is-retraction-inv-concat-right-identification-coherence-square-identifications = + is-retraction-inv-concat' (ap (concat top _) s) +``` + +We record that this construction is an equivalence in +[`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md). + +#### Concatenating identifications of the bottom edge with a coherence of a commuting square of identifications + +Consider a commuting diagram of identifications + +```text + top + x -------> y + | | + left | | right + ∨ bottom ∨ + z -------> w. + -------> + bottom' +``` + +with an identification `bottom ＝ bottom'`. Then we get an equivalence + +```text + top top + x -------> y x -------> y + | | | | + left | | right ≃ left | | right + ∨ ∨ ∨ ∨ + z -------> w z -------> w. + bottom bottom' +``` + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) + {bottom' : z ＝ w} (s : bottom ＝ bottom') + where + + concat-bottom-identification-coherence-square-identifications : + coherence-square-identifications top left right bottom → + coherence-square-identifications top left right bottom' + concat-bottom-identification-coherence-square-identifications t = + inv (ap (concat left _) s) ∙ t + + inv-concat-bottom-identification-coherence-square-identifications : + coherence-square-identifications top left right bottom' → + coherence-square-identifications top left right bottom + inv-concat-bottom-identification-coherence-square-identifications t = + ap (concat left _) s ∙ t + + is-section-inv-concat-bottom-identification-coherence-square-identifications : + is-section + concat-bottom-identification-coherence-square-identifications + inv-concat-bottom-identification-coherence-square-identifications + is-section-inv-concat-bottom-identification-coherence-square-identifications = + is-retraction-inv-concat (ap (concat left _) s) + + is-retraction-inv-concat-bottom-identification-coherence-square-identifications : + is-retraction + concat-bottom-identification-coherence-square-identifications + inv-concat-bottom-identification-coherence-square-identifications + is-retraction-inv-concat-bottom-identification-coherence-square-identifications = + is-section-inv-concat (ap (concat left _) s) +``` + +We record that this construction is an equivalence in +[`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md). + +### Whiskering and splicing coherences of commuting squares of identifications + +Given a commuting square of identifications + +```text + top + x -------> y + | | + left | | right + ∨ ∨ + z -------> w, + bottom +``` + +we may consider four ways of attaching new identifications to it: + +1. Prepending `p : u ＝ x` to the left gives us a commuting square + + ```text + p ∙ top + u -------> y + | | + p ∙ left | | right + ∨ ∨ + z -------> w. + bottom + ``` + + More precisely, we have an equivalence + + ```text + (left ∙ bottom ＝ top ∙ right) ≃ ((p ∙ left) ∙ bottom ＝ (p ∙ top) ∙ right). + ``` + +2. Appending an identification `p : w ＝ u` to the right gives a commuting + square of identifications + + ```text + top + x ------------> y + | | + left | | right ∙ p + ∨ ∨ + z ------------> u. + bottom ∙ p + ``` + + More precisely, we have an equivalence + + ```text + (left ∙ bottom ＝ top ∙ right) ≃ (left ∙ (bottom ∙ p) ＝ top ∙ (right ∙ p)). + ``` + +3. Splicing an identification `p : z ＝ u` and its inverse into the middle gives + a commuting square of identifications + + ```text + top + x --------------> y + | | + left ∙ p | | right + ∨ ∨ + u --------------> w. + p⁻¹ ∙ bottom + ``` + + More precisely, we have an equivalence + + ```text + (left ∙ bottom ＝ top ∙ right) ≃ ((left ∙ p) ∙ (p⁻¹ ∙ bottom) ＝ top ∙ right). + ``` + + Similarly, we have an equivalence + + ```text + (left ∙ bottom ＝ top ∙ right) ≃ ((left ∙ p⁻¹) ∙ (p ∙ bottom) ＝ top ∙ right). + ``` + +4. Splicing an identification `p : y ＝ u` and its inverse into the middle gives + a commuting square of identifications + + ```text + top ∙ p + x --------> u + | | + left | | p⁻¹ ∙ right + ∨ ∨ + z --------> w. + bottom + ``` + + More precisely, we have an equivalence + + ```text + (left ∙ bottom ＝ top ∙ right) ≃ (left ∙ bottom ＝ (top ∙ p) ∙ (p⁻¹ ∙ right)). + ``` + + Similarly, we have an equivalence + + ```text + (left ∙ bottom ＝ top ∙ right) ≃ (left ∙ bottom ＝ (top ∙ p⁻¹) ∙ (p ∙ right)). + ``` + +These operations are useful in proofs involving path algebra, because taking +`equiv-right-whisker-concat-coherence-square-identifications` as an example, it +provides us with two maps: the forward direction states +`(p ∙ r ＝ q ∙ s) → (p ∙ (r ∙ t)) ＝ q ∙ (s ∙ t))`, which allows one to append +an identification without needing to reassociate on the right, and the backwards +direction conversely allows one to cancel out an identification in parentheses. + +#### Left whiskering coherences of commuting squares of identifications + +For any identification `p : u ＝ x` we obtain an equivalence + +```text + top p ∙ top + x -------> y u -------> y + | | | | + left | | right ≃ p ∙ left | | right + ∨ ∨ ∨ ∨ + z -------> w z -------> w + bottom bottom +``` + +of coherences of commuting squares of identifications. + +```agda +module _ + {l : Level} {A : UU l} {x y z w u : A} + where + + left-whisker-concat-coherence-square-identifications : + (p : u ＝ x) + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → + coherence-square-identifications top left right bottom → + coherence-square-identifications (p ∙ top) (p ∙ left) right bottom + left-whisker-concat-coherence-square-identifications + refl top left right bottom = + id + + left-unwhisker-concat-coherence-square-identifications : + (p : u ＝ x) + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → + coherence-square-identifications (p ∙ top) (p ∙ left) right bottom → + coherence-square-identifications top left right bottom + left-unwhisker-concat-coherence-square-identifications + refl top left right bottom = + id +``` + +#### Right whiskering coherences of commuting squares of identifications + +For any identification `p : w ＝ u` we obtain an equivalence + +```text + top top + x -------> y x ------------> y + | | | | + left | | right ≃ left | | right ∙ p + ∨ ∨ ∨ ∨ + z -------> w z ------------> w + bottom bottom ∙ p +``` + +of coherences of commuting squares of identifications. + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) + where + + right-whisker-concat-coherence-square-identifications : + coherence-square-identifications top left right bottom → + {u : A} (p : w ＝ u) → + coherence-square-identifications top left (right ∙ p) (bottom ∙ p) + right-whisker-concat-coherence-square-identifications s refl = + concat-bottom-identification-coherence-square-identifications + ( top) + ( left) + ( right ∙ refl) + ( bottom) + ( inv right-unit) + ( concat-right-identification-coherence-square-identifications + ( top) + ( left) + ( right) + ( bottom) + ( inv right-unit) + ( s)) + + right-unwhisker-cohernece-square-identifications : + {u : A} (p : w ＝ u) → + coherence-square-identifications top left (right ∙ p) (bottom ∙ p) → + coherence-square-identifications top left right bottom + right-unwhisker-cohernece-square-identifications refl = + ( inv-concat-right-identification-coherence-square-identifications + ( top) + ( left) + ( right) + ( bottom) + ( inv right-unit)) ∘ + ( inv-concat-bottom-identification-coherence-square-identifications + ( top) + ( left) + ( right ∙ refl) + ( bottom) + ( inv right-unit)) +``` + +### Double whiskering of commuting squares of identifications + +```agda +module _ + {l : Level} {A : UU l} {x y z u v w : A} + where + + double-whisker-coherence-square-identifications : + (p : x ＝ y) + (top : y ＝ u) (left : y ＝ z) (right : u ＝ v) (bottom : z ＝ v) + (s : v ＝ w) → + coherence-square-identifications top left right bottom → + coherence-square-identifications + ( p ∙ top) + ( p ∙ left) + ( right ∙ s) + ( bottom ∙ s) + double-whisker-coherence-square-identifications + p top left right bottom q H = + left-whisker-concat-coherence-square-identifications p top left + ( right ∙ q) + ( bottom ∙ q) + ( right-whisker-concat-coherence-square-identifications + ( top) + ( left) + ( right) + ( bottom) + ( H) + ( q)) +``` + +#### Left splicing coherences of commuting squares of identifications + +For any inverse pair of identifications `p : y ＝ u` and `q : u ＝ y` equipped +with `α : inv p ＝ q` we obtain an equivalence + +```text + top top + x -------> y x -----------> y + | | | | + left | | right ≃ left ∙ p | | right + ∨ ∨ ∨ ∨ + z -------> w u -----------> w + bottom q ∙ bottom +``` + +of coherences of commuting squares of identifications. + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) + where + + left-splice-coherence-square-identifications : + {u : A} (p : z ＝ u) (q : u ＝ z) (α : inv p ＝ q) → + coherence-square-identifications top left right bottom → + coherence-square-identifications top (left ∙ p) right (q ∙ bottom) + left-splice-coherence-square-identifications refl .refl refl = + concat-left-identification-coherence-square-identifications + ( top) + ( left) + ( right) + ( bottom) + ( inv right-unit) + + left-unsplice-coherence-square-identifications : + {u : A} (p : z ＝ u) (q : u ＝ z) (α : inv p ＝ q) → + coherence-square-identifications top (left ∙ p) right (q ∙ bottom) → + coherence-square-identifications top left right bottom + left-unsplice-coherence-square-identifications refl .refl refl = + inv-concat-left-identification-coherence-square-identifications + ( top) + ( left) + ( right) + ( bottom) + ( inv right-unit) +``` + +#### Right splicing coherences of commuting squares of identifications + +For any inverse pair of identifications `p : y ＝ u` and `q : u ＝ y` equipped +with `α : inv p ＝ q` we obtain an equivalence + +```text + top top ∙ p + x -------> y x --------> u + | | | | + left | | right ≃ left | | q ∙ right + ∨ ∨ ∨ ∨ + z -------> w z --------> w + bottom bottom +``` + +of coherences of commuting squares of identifications. + +```agda +module _ + {l : Level} {A : UU l} {x y z w : A} + (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) + where + + right-splice-coherence-square-identifications : + {u : A} (p : y ＝ u) (q : u ＝ y) (α : inv p ＝ q) → + coherence-square-identifications top left right bottom → + coherence-square-identifications (top ∙ p) left (inv p ∙ right) bottom + right-splice-coherence-square-identifications refl .refl refl = + concat-top-identification-coherence-square-identifications + ( top) + ( left) + ( right) + ( bottom) + ( inv right-unit) + + right-unsplice-coherence-square-identifications : + {u : A} (p : y ＝ u) (q : u ＝ y) (α : inv p ＝ q) → + coherence-square-identifications (top ∙ p) left (inv p ∙ right) bottom → + coherence-square-identifications top left right bottom + right-unsplice-coherence-square-identifications refl .refl refl = + inv-concat-top-identification-coherence-square-identifications + ( top) + ( left) + ( right) + ( bottom) + ( inv right-unit) +``` + +### Horizontally pasting squares of identifications + +Consider two squares of identifications as in the diagram + +```text + top-left top-right + a -------------> b -------------> c + | | | + left | | middle | right + ∨ ∨ ∨ + d -------------> e -------------> f + bottom-left bottom-right +``` + +with `s : left ∙ bottom-left ＝ top-left ∙ middle` and +`t : middle ∙ bottom-right ＝ top-right ∙ right`. Then the outer square +commutes. + +```agda +module _ + {l : Level} {A : UU l} {a b c d e f : A} + (top-left : a ＝ b) (top-right : b ＝ c) + (left : a ＝ d) (middle : b ＝ e) (right : c ＝ f) + (bottom-left : d ＝ e) (bottom-right : e ＝ f) + where + + horizontal-pasting-coherence-square-identifications : + coherence-square-identifications top-left left middle bottom-left → + coherence-square-identifications top-right middle right bottom-right → + coherence-square-identifications + (top-left ∙ top-right) left right (bottom-left ∙ bottom-right) + horizontal-pasting-coherence-square-identifications s t = + ( right-whisker-concat-coherence-square-identifications + ( top-left) + ( left) + ( middle) + ( bottom-left) + ( s) + ( bottom-right)) ∙ + ( ( inv (assoc top-left middle bottom-right)) ∙ + ( left-whisker-concat-coherence-square-identifications + ( top-left) + ( top-right) + ( middle) + ( right) + ( bottom-right) + ( t))) +``` + +### Vertically pasting squares of identifications + +Consider two squares of identifications as in the diagram + +```text + top + a --------> b + | | + top-left | | top-right + ∨ middle ∨ + c --------> d + | | + bottom-left | | bottom-right + ∨ ∨ + e --------> f + bottom +``` + +with `s : top-left ∙ middle ＝ top ∙ top-right` and +`t : bottom-left ∙ bottom ＝ middle ∙ bottom-right`. Then the outer square +commutes. + +```agda +module _ + {l : Level} {A : UU l} {a b c d e f : A} + (top : a ＝ b) (top-left : a ＝ c) (top-right : b ＝ d) + (middle : c ＝ d) (bottom-left : c ＝ e) (bottom-right : d ＝ f) + (bottom : e ＝ f) + where + + vertical-pasting-coherence-square-identifications : + coherence-square-identifications top top-left top-right middle → + coherence-square-identifications middle bottom-left bottom-right bottom → + coherence-square-identifications + top (top-left ∙ bottom-left) (top-right ∙ bottom-right) bottom + vertical-pasting-coherence-square-identifications s t = + ( left-whisker-concat-coherence-square-identifications + ( top-left) + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( t)) ∙ + ( ( assoc top-left middle bottom-right) ∙ + ( right-whisker-concat-coherence-square-identifications + ( top) + ( top-left) + ( top-right) + ( middle) + ( s) + ( bottom-right))) +``` + +## Properties + +### Left unit law for horizontal pasting of commuting squares of identifications + +```agda +module _ + {l : Level} {A : UU l} {a b c d : A} + where + + left-unit-law-horizontal-pasting-coherence-square-identifications : + (top : a ＝ b) (left : a ＝ c) (right : b ＝ d) (bottom : c ＝ d) + (s : coherence-square-identifications top left right bottom) → + horizontal-pasting-coherence-square-identifications + ( refl) + ( top) + ( left) + ( left) + ( right) + ( refl) + ( bottom) + ( horizontal-refl-coherence-square-identifications left) + ( s) ＝ + s + left-unit-law-horizontal-pasting-coherence-square-identifications + refl refl right refl s = refl +``` + +### Right unit law for horizontal pasting of commuting squares of identifications + +```agda +module _ + {l : Level} {A : UU l} {a b c d : A} + where + + right-unit-law-horizontal-pasting-coherence-square-identifications : + (top : a ＝ b) (left : a ＝ c) (right : b ＝ d) (bottom : c ＝ d) + (s : coherence-square-identifications top left right bottom) → + horizontal-pasting-coherence-square-identifications + ( top) + ( refl) + ( left) + ( right) + ( right) + ( bottom) + ( refl) + ( s) + ( horizontal-refl-coherence-square-identifications right) ∙ + right-whisker-concat right-unit right ＝ + left-whisker-concat left right-unit ∙ s + right-unit-law-horizontal-pasting-coherence-square-identifications + refl refl .refl refl refl = + refl +``` + +### Left unit law for vertical pasting of commuting squares of identifications + +```agda +module _ + {l : Level} {A : UU l} {a b c d : A} + where + + left-unit-law-vertical-pasting-coherence-square-identifications : + (top : a ＝ b) (left : a ＝ c) (right : b ＝ d) (bottom : c ＝ d) + (s : coherence-square-identifications top left right bottom) → + vertical-pasting-coherence-square-identifications + ( top) + ( refl) + ( refl) + ( top) + ( left) + ( right) + ( bottom) + ( vertical-refl-coherence-square-identifications top) + ( s) ＝ + s + left-unit-law-vertical-pasting-coherence-square-identifications + refl refl .refl refl refl = refl +``` + +### Right unit law for vertical pasting of commuting squares of identifications + +```agda +module _ + {l : Level} {A : UU l} {a b c d : A} + where + + right-unit-law-vertical-pasting-coherence-square-identifications : + (top : a ＝ b) (left : a ＝ c) (right : b ＝ d) (bottom : c ＝ d) + (s : coherence-square-identifications top left right bottom) → + vertical-pasting-coherence-square-identifications + ( top) + ( left) + ( right) + ( bottom) + ( refl) + ( refl) + ( bottom) + ( s) + ( vertical-refl-coherence-square-identifications bottom) ∙ + left-whisker-concat top right-unit ＝ + right-whisker-concat right-unit bottom ∙ s + right-unit-law-vertical-pasting-coherence-square-identifications + refl refl .(refl ∙ refl) refl refl = + refl +``` + +### Computing the right whiskering of a vertically constant square with an identification + +Consider the vertically constant square of identifications + +```text + p + x -----> y + | | + refl | | refl + ∨ ∨ + x -----> y + p +``` + +at an identification `p : x ＝ y`, and consider an identification `q : y ＝ z`. +Then the right whiskering of the above square with `q` is the commuting square +of identifications + +```text + p + x -------> y + | | + refl | refl | q + ∨ ∨ + x -------> z + p ∙ q +``` + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + right-whisker-concat-vertical-refl-coherence-square-identifications : + {x y z : A} (p : x ＝ y) (q : y ＝ z) → + right-whisker-concat-coherence-square-identifications p refl refl p + ( vertical-refl-coherence-square-identifications p) + ( q) ＝ + refl + right-whisker-concat-vertical-refl-coherence-square-identifications + refl refl = + refl +``` + +### Computing the right whiskering of a horizontally constant square with an identification + +Consider a horizontally constant commuting square of identifications + +```text + refl + x -----> x + | | + p | | p + ∨ ∨ + y -----> y + refl +``` + +at an identification `p` and consider an identification `q : y ＝ z`. Then the +right whiskering of the above square with `q` is the square + +```text + refl + x -----> x + | | + p | refl | p ∙ q + ∨ ∨ + y -----> z. + q +``` + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + right-whisker-concat-horizontal-refl-coherence-square-identifications : + {x y z : A} (p : x ＝ y) (q : y ＝ z) → + right-whisker-concat-coherence-square-identifications refl p p refl + ( horizontal-refl-coherence-square-identifications p) + ( q) ＝ + refl + right-whisker-concat-horizontal-refl-coherence-square-identifications + refl refl = + refl +``` + +### Computing the left whiskering of a horizontally constant square with an identification + +Consider an identification `p : x ＝ y` and a horizontally constant commuting +square of identifications + +```text + refl + y -----> y + | | + q | | q + ∨ ∨ + z -----> z + refl +``` + +at an identification `q : y ＝ z`. The the left whiskering of the above square +with `p` is the commuting square + +```text + q ∙ refl + x ------------------------------------------------------> y + | | + q ∙ p | right-unit ∙ inv (right-whisker-concat right-unit p) | p + ∨ ∨ + z ------------------------------------------------------> z. + refl +``` + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + left-whisker-concat-horizontal-refl-coherence-square-identifications : + {x y z : A} (p : x ＝ y) (q : y ＝ z) → + left-whisker-concat-coherence-square-identifications p refl q q refl + ( horizontal-refl-coherence-square-identifications q) ∙ + right-whisker-concat right-unit q ＝ + right-unit + left-whisker-concat-horizontal-refl-coherence-square-identifications + refl refl = + refl + + left-whisker-concat-horizontal-refl-coherence-square-identifications' : + {x y z : A} (p : x ＝ y) (q : y ＝ z) → + left-whisker-concat-coherence-square-identifications p refl q q refl + ( horizontal-refl-coherence-square-identifications q) ＝ + right-unit ∙ inv (right-whisker-concat right-unit q) + left-whisker-concat-horizontal-refl-coherence-square-identifications' + refl refl = + refl +``` + +### Computing the left whiskering of a vertically constant square with an identification + +Consider the vertically constant square of identifications + +```text + q + y -----> z + | | + refl | | refl + ∨ ∨ + y -----> z + q +``` + +at an identification `q : y ＝ z` and consider an identification `p : x ＝ y`. +Then the left whiskering of the above square with `p` is the square + +```text + p ∙ q + x ---------------------------------------------------> z + | | + p ∙ refl | right-whisker-concat right-unit q ∙ inv right-unit | refl + ∨ ∨ + y ---------------------------------------------------> z. + q +``` + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + left-whisker-concat-vertical-refl-coherence-square-identifications : + {x y z : A} (p : x ＝ y) (q : y ＝ z) → + left-whisker-concat-coherence-square-identifications p q refl refl q + ( vertical-refl-coherence-square-identifications q) ∙ + right-unit ＝ + right-whisker-concat right-unit q + left-whisker-concat-vertical-refl-coherence-square-identifications + refl refl = + refl + + left-whisker-concat-vertical-refl-coherence-square-identifications' : + {x y z : A} (p : x ＝ y) (q : y ＝ z) → + left-whisker-concat-coherence-square-identifications p q refl refl q + ( vertical-refl-coherence-square-identifications q) ＝ + right-whisker-concat right-unit q ∙ inv right-unit + left-whisker-concat-vertical-refl-coherence-square-identifications' + refl refl = + refl +``` + +### Left whiskering horizontal concatenations of squares with identifications + +Consider a commuting diagram of identifications of the form + +```text + top-left top-right + a -------------> c -------------> e + | | | + left | | middle | right + ∨ ∨ ∨ + b -------------> d -------------> f + bottom-left bottom-right +``` + +and consider an identification `p : x ＝ a`. Then the left whiskering of `p` and +the horizontal concatenation of coherences of commuting squares is up to +associativity the horizontal concatenation of the squares + +```text + p ∙ top-left top-right + x -------------> c -------------> e + | | | + p ∙ left | | middle | right + ∨ ∨ ∨ + b -------------> d -------------> f + bottom-left bottom-right +``` + +where the left square is the left whiskering of `p` and the original left +square. + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + left-whisker-concat-horizontal-pasting-coherence-square-identifications : + {x a b c d e f : A} (p : x ＝ a) + (top-left : a ＝ c) (top-right : c ＝ e) + (left : a ＝ b) (middle : c ＝ d) (right : e ＝ f) + (bottom-left : b ＝ d) (bottom-right : d ＝ f) + (l : coherence-square-identifications top-left left middle bottom-left) + (r : coherence-square-identifications top-right middle right bottom-right) → + left-whisker-concat-coherence-square-identifications p + ( top-left ∙ top-right) + ( left) + ( right) + ( bottom-left ∙ bottom-right) + ( horizontal-pasting-coherence-square-identifications + ( top-left) + ( top-right) + ( left) + ( middle) + ( right) + ( bottom-left) + ( bottom-right) + ( l) + ( r)) ＝ + horizontal-pasting-coherence-square-identifications + ( p ∙ top-left) + ( top-right) + ( p ∙ left) + ( middle) + ( right) + ( bottom-left) + ( bottom-right) + ( left-whisker-concat-coherence-square-identifications p + ( top-left) + ( left) + ( middle) + ( bottom-left) + ( l)) + ( r) ∙ + right-whisker-concat + ( assoc p top-left top-right) + ( right) + left-whisker-concat-horizontal-pasting-coherence-square-identifications + refl top-left top-right left middle right bottom-left bottom-right l r = + inv right-unit +``` + +### Left whiskering vertical concatenations of squares with identifications + +Consider two squares of identifications as in the diagram + +```text + top + a --------> b + | | + top-left | | top-right + ∨ middle ∨ + c --------> d + | | + bottom-left | | bottom-right + ∨ ∨ + e --------> f + bottom +``` + +and consider an identification `p : x ＝ a`. Then the left whiskering of `p` +with the vertical pasting of the two squares above is up to associativity the +vertical pasting of the squares + +```text + p ∙ top + x --------> b + | | + p ∙ top-left | | top-right + ∨ middle ∨ + c --------> d + | | + bottom-left | | bottom-right + ∨ ∨ + e --------> f. + bottom +``` + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + left-whisker-concat-vertical-concat-coherence-square-identifications : + {x a b c d e f : A} (p : x ＝ a) → + (top : a ＝ b) (top-left : a ＝ c) (top-right : b ＝ d) (middle : c ＝ d) + (bottom-left : c ＝ e) (bottom-right : d ＝ f) (bottom : e ＝ f) + (t : coherence-square-identifications top top-left top-right middle) → + (b : + coherence-square-identifications middle bottom-left bottom-right bottom) → + right-whisker-concat (assoc p top-left bottom-left) bottom ∙ + left-whisker-concat-coherence-square-identifications p + ( top) + ( top-left ∙ bottom-left) + ( top-right ∙ bottom-right) + ( bottom) + ( vertical-pasting-coherence-square-identifications + ( top) + ( top-left) + ( top-right) + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( t) + ( b)) ＝ + vertical-pasting-coherence-square-identifications + ( p ∙ top) + ( p ∙ top-left) + ( top-right) + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( left-whisker-concat-coherence-square-identifications p + ( top) + ( top-left) + ( top-right) + ( middle) + ( t)) + ( b) + left-whisker-concat-vertical-concat-coherence-square-identifications + refl top top-left top-right middle bottom-left bottom-right bottom t b = + refl +``` + +### Right whiskering horizontal pastings of commuting squares of identifications + +Consider a commuting diagram of identifications of the form + +```text + top-left top-right + a -------------> c -------------> e + | | | + left | | middle | right + ∨ ∨ ∨ + b -------------> d -------------> f + bottom-left bottom-right +``` + +and consider an identification `q : f ＝ y`. Then the right whiskering of the +horizontal pasting of the squares above is up to associativity the horizontal +pasting of the squares + +```text + top-left top-right + a -------------> c ------------------> e + | | | + left | | middle | right ∙ q + ∨ ∨ ∨ + b -------------> d ------------------> y + bottom-left bottom-right ∙ q +``` + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + right-whisker-concat-horizontal-pasting-coherence-square-identifications : + {a b c d e f y : A} + (top-left : a ＝ c) (top-right : c ＝ e) + (left : a ＝ b) (middle : c ＝ d) (right : e ＝ f) + (bottom-left : b ＝ d) (bottom-right : d ＝ f) + (l : coherence-square-identifications top-left left middle bottom-left) → + (r : coherence-square-identifications top-right middle right bottom-right) → + (q : f ＝ y) → + right-whisker-concat-coherence-square-identifications + ( top-left ∙ top-right) + ( left) + ( right) + ( bottom-left ∙ bottom-right) + ( horizontal-pasting-coherence-square-identifications + ( top-left) + ( top-right) + ( left) + ( middle) + ( right) + ( bottom-left) + ( bottom-right) + ( l) + ( r)) + ( q) ＝ + left-whisker-concat left (assoc bottom-left bottom-right q) ∙ + horizontal-pasting-coherence-square-identifications + ( top-left) + ( top-right) + ( left) + ( middle) + ( right ∙ q) + ( bottom-left) + ( bottom-right ∙ q) + ( l) + ( right-whisker-concat-coherence-square-identifications + ( top-right) + ( middle) + ( right) + ( bottom-right) + ( r) + ( q)) + right-whisker-concat-horizontal-pasting-coherence-square-identifications + refl refl refl .refl .refl refl refl refl refl refl = + refl +``` + +### Right whiskering vertical concatenations of squares with identifications + +Consider two squares of identifications as in the diagram + +```text + top + a --------> b + | | + top-left | | top-right + ∨ middle ∨ + c --------> d + | | + bottom-left | | bottom-right + ∨ ∨ + e --------> f + bottom +``` + +and consider an identification `q : f ＝ y`. Then the right whiskering of the +vertical pasting of the two squares above with `q` is up to associativity the +vertical pasting of the squares + +```text + top + a ------------> b + | | + top-left | | top-right + ∨ middle ∨ + c ------------> d + | | + bottom-left | | bottom-right ∙ q + ∨ ∨ + e ------------> y. + bottom ∙ q +``` + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + right-whisker-concat-vertical-pasting-coherence-square-identifications : + {a b c d e f y : A} + (top : a ＝ b) (top-left : a ＝ c) (top-right : b ＝ d) + (middle : c ＝ d) + (bottom-left : c ＝ e) (bottom-right : d ＝ f) (bottom : e ＝ f) + (t : coherence-square-identifications top top-left top-right middle) → + (b : + coherence-square-identifications middle bottom-left bottom-right bottom) → + (q : f ＝ y) → + right-whisker-concat-coherence-square-identifications + ( top) + ( top-left ∙ bottom-left) + ( top-right ∙ bottom-right) + ( bottom) + ( vertical-pasting-coherence-square-identifications + ( top) + ( top-left) + ( top-right) + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( t) + ( b)) + ( q) ∙ + left-whisker-concat top (assoc top-right bottom-right q) ＝ + vertical-pasting-coherence-square-identifications + ( top) + ( top-left) + ( top-right) + ( middle) + ( bottom-left) + ( bottom-right ∙ q) + ( bottom ∙ q) + ( t) + ( right-whisker-concat-coherence-square-identifications + ( middle) + ( bottom-left) + ( bottom-right) + ( bottom) + ( b) + ( q)) + right-whisker-concat-vertical-pasting-coherence-square-identifications + refl refl .refl refl refl .refl refl refl refl refl = + refl +``` diff --git a/src/foundation-core/commuting-squares-of-maps.lagda.md b/src/foundation-core/commuting-squares-of-maps.lagda.md index 13c4b21118..92e9ba73f6 100644 --- a/src/foundation-core/commuting-squares-of-maps.lagda.md +++ b/src/foundation-core/commuting-squares-of-maps.lagda.md @@ -234,23 +234,23 @@ If the horizontal/vertical maps in a commuting square are both commuting if we invert those equivalences. ```agda -coherence-square-maps-inv-equiv-horizontal : +horizontal-inv-equiv-coherence-square-maps : {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} (top : A ≃ B) (left : A → X) (right : B → Y) (bottom : X ≃ Y) → coherence-square-maps (map-equiv top) left right (map-equiv bottom) → coherence-square-maps (map-inv-equiv top) right left (map-inv-equiv bottom) -coherence-square-maps-inv-equiv-horizontal top left right bottom H b = +horizontal-inv-equiv-coherence-square-maps top left right bottom H b = map-eq-transpose-equiv-inv ( bottom) ( ( ap right (inv (is-section-map-inv-equiv top b))) ∙ ( inv (H (map-inv-equiv top b)))) -coherence-square-maps-inv-equiv-vertical : +vertical-inv-equiv-coherence-square-maps : {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} (top : A → B) (left : A ≃ X) (right : B ≃ Y) (bottom : X → Y) → coherence-square-maps top (map-equiv left) (map-equiv right) bottom → coherence-square-maps bottom (map-inv-equiv left) (map-inv-equiv right) top -coherence-square-maps-inv-equiv-vertical top left right bottom H x = +vertical-inv-equiv-coherence-square-maps top left right bottom H x = map-eq-transpose-equiv ( right) ( ( inv (H (map-inv-equiv left x))) ∙ @@ -270,12 +270,12 @@ coherence-square-maps-inv-equiv : ( map-inv-equiv left) ( map-inv-equiv top) coherence-square-maps-inv-equiv top left right bottom H = - coherence-square-maps-inv-equiv-vertical + vertical-inv-equiv-coherence-square-maps ( map-inv-equiv top) ( right) ( left) ( map-inv-equiv bottom) - ( coherence-square-maps-inv-equiv-horizontal + ( horizontal-inv-equiv-coherence-square-maps ( top) ( map-equiv left) ( map-equiv right) diff --git a/src/foundation-core/contractible-maps.lagda.md b/src/foundation-core/contractible-maps.lagda.md index 46a3cecce5..624c850cd4 100644 --- a/src/foundation-core/contractible-maps.lagda.md +++ b/src/foundation-core/contractible-maps.lagda.md @@ -66,10 +66,9 @@ module _ ( ( inv ( contraction ( H (f x)) - ( pair - ( map-inv-is-contr-map H (f x)) + ( ( map-inv-is-contr-map H (f x)) , ( is-section-map-inv-is-contr-map H (f x))))) ∙ - ( contraction (H (f x)) (pair x refl))) + ( contraction (H (f x)) (x , refl))) abstract is-equiv-is-contr-map : is-contr-map f → is-equiv f @@ -93,14 +92,14 @@ module _ pr1 (center-fiber-is-coherently-invertible H y) = map-inv-is-coherently-invertible H y pr2 (center-fiber-is-coherently-invertible H y) = - is-retraction-is-coherently-invertible H y + is-section-map-inv-is-coherently-invertible H y contraction-fiber-is-coherently-invertible : (H : is-coherently-invertible f) → (y : B) → (t : fiber f y) → (center-fiber-is-coherently-invertible H y) ＝ t - contraction-fiber-is-coherently-invertible H y (pair x refl) = + contraction-fiber-is-coherently-invertible H y (x , refl) = eq-Eq-fiber f y - ( is-section-is-coherently-invertible H x) + ( is-retraction-map-inv-is-coherently-invertible H x) ( ( right-unit) ∙ ( inv ( coh-is-coherently-invertible H x))) diff --git a/src/foundation-core/equivalences.lagda.md b/src/foundation-core/equivalences.lagda.md index 3d1af2c9bb..843b2ae090 100644 --- a/src/foundation-core/equivalences.lagda.md +++ b/src/foundation-core/equivalences.lagda.md @@ -7,7 +7,6 @@ module foundation-core.equivalences where
Imports ```agda -open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.universe-levels @@ -181,18 +180,17 @@ module _ pr2 (pr2 (is-equiv-is-invertible' (g , H , K))) = K is-equiv-is-invertible : - (g : B → A) (H : (f ∘ g) ~ id) (K : (g ∘ f) ~ id) → is-equiv f + (g : B → A) (H : f ∘ g ~ id) (K : g ∘ f ~ id) → is-equiv f is-equiv-is-invertible g H K = is-equiv-is-invertible' (g , H , K) is-retraction-map-section-is-equiv : (H : is-equiv f) → is-retraction f (map-section-is-equiv H) is-retraction-map-section-is-equiv H = - ( ( ( inv-htpy - ( ( is-retraction-map-retraction-is-equiv H) ·r - ( map-section-is-equiv H))) ∙h - ( map-retraction-is-equiv H ·l is-section-map-section-is-equiv H)) ·r - ( f)) ∙h + ( ( inv-htpy + ( ( is-retraction-map-retraction-is-equiv H) ·r + ( map-section-is-equiv H ∘ f))) ∙h + ( map-retraction-is-equiv H ·l is-section-map-section-is-equiv H ·r f)) ∙h ( is-retraction-map-retraction-is-equiv H) is-invertible-is-equiv : is-equiv f → is-invertible f @@ -209,15 +207,26 @@ module _ where abstract - is-coherently-invertible-is-equiv : is-equiv f → is-coherently-invertible f + is-coherently-invertible-is-equiv : + is-equiv f → is-coherently-invertible f is-coherently-invertible-is-equiv = is-coherently-invertible-is-invertible ∘ is-invertible-is-equiv - abstract is-equiv-is-coherently-invertible : is-coherently-invertible f → is-equiv f is-equiv-is-coherently-invertible H = is-equiv-is-invertible' (is-invertible-is-coherently-invertible H) + + is-transpose-coherently-invertible-is-equiv : + is-equiv f → is-transpose-coherently-invertible f + is-transpose-coherently-invertible-is-equiv = + is-transpose-coherently-invertible-is-invertible ∘ is-invertible-is-equiv + + is-equiv-is-transpose-coherently-invertible : + is-transpose-coherently-invertible f → is-equiv f + is-equiv-is-transpose-coherently-invertible H = + is-equiv-is-invertible' + ( is-invertible-is-transpose-coherently-invertible H) ``` ### Structure obtained from being coherently invertible @@ -232,11 +241,13 @@ module _ is-section-map-inv-is-equiv : is-section f map-inv-is-equiv is-section-map-inv-is-equiv = - is-section-map-inv-is-invertible (is-invertible-is-equiv H) + is-section-map-inv-is-coherently-invertible-is-invertible + ( is-invertible-is-equiv H) is-retraction-map-inv-is-equiv : is-retraction f map-inv-is-equiv is-retraction-map-inv-is-equiv = - is-retraction-map-inv-is-invertible (is-invertible-is-equiv H) + is-retraction-map-inv-is-coherently-invertible-is-invertible + ( is-invertible-is-equiv H) coherence-map-inv-is-equiv : coherence-is-coherently-invertible f @@ -244,7 +255,7 @@ module _ ( is-section-map-inv-is-equiv) ( is-retraction-map-inv-is-equiv) coherence-map-inv-is-equiv = - coherence-map-inv-is-invertible (is-invertible-is-equiv H) + coh-is-coherently-invertible-is-invertible (is-invertible-is-equiv H) is-equiv-map-inv-is-equiv : is-equiv map-inv-is-equiv is-equiv-map-inv-is-equiv = @@ -415,12 +426,12 @@ module _ pr2 (is-equiv-comp g h (sh , rh) (sg , rg)) = retraction-comp g h rg rh - equiv-comp : (B ≃ X) → (A ≃ B) → (A ≃ X) - pr1 (equiv-comp g h) = (map-equiv g) ∘ (map-equiv h) + equiv-comp : B ≃ X → A ≃ B → A ≃ X + pr1 (equiv-comp g h) = map-equiv g ∘ map-equiv h pr2 (equiv-comp g h) = is-equiv-comp (pr1 g) (pr1 h) (pr2 h) (pr2 g) infixr 15 _∘e_ - _∘e_ : (B ≃ X) → (A ≃ B) → (A ≃ X) + _∘e_ : B ≃ X → A ≃ B → A ≃ X _∘e_ = equiv-comp ``` @@ -483,14 +494,12 @@ module _ is-equiv-htpy' (map-equiv e) H (is-equiv-map-equiv e) htpy-map-inv-is-equiv : - {f g : A → B} (G : f ~ g) (H : is-equiv f) (K : is-equiv g) → - map-inv-is-equiv H ~ map-inv-is-equiv K - htpy-map-inv-is-equiv G H K b = - ( inv - ( is-retraction-map-inv-is-equiv K (map-inv-is-equiv H b))) ∙ - ( ap (map-inv-is-equiv K) - ( ( inv (G (map-inv-is-equiv H b))) ∙ - ( is-section-map-inv-is-equiv H b))) + {f g : A → B} (H : f ~ g) (F : is-equiv f) (G : is-equiv g) → + map-inv-is-equiv F ~ map-inv-is-equiv G + htpy-map-inv-is-equiv H F G = + htpy-map-inv-is-invertible H + ( is-invertible-is-equiv F) + ( is-invertible-is-equiv G) ``` ### Any retraction of an equivalence is an equivalence @@ -523,16 +532,22 @@ module _ where abstract - is-equiv-section-is-equiv : - ( section-f : section f) → is-equiv (pr1 section-f) → is-equiv f - is-equiv-section-is-equiv (g , is-section-g) is-equiv-section-f = - is-equiv-htpy h - ( ( f ·l (inv-htpy (is-section-map-inv-is-equiv is-equiv-section-f))) ∙h - ( right-whisker-comp is-section-g h)) - ( is-equiv-map-inv-is-equiv is-equiv-section-f) - where - h : A → B - h = map-inv-is-equiv is-equiv-section-f + is-equiv-is-equiv-section : + (s : section f) → is-equiv (map-section f s) → is-equiv f + is-equiv-is-equiv-section (g , G) S = is-equiv-is-retraction S G +``` + +### If a retraction of `f` is an equivalence, then `f` is an equivalence + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) + where + + abstract + is-equiv-is-equiv-retraction : + (r : retraction f) → is-equiv (map-retraction f r) → is-equiv f + is-equiv-is-equiv-retraction (g , G) R = is-equiv-is-section R G ``` ### Any section of an equivalence is homotopic to its inverse @@ -662,38 +677,10 @@ module _ abstract is-emb-is-equiv : {f : A → B} → is-equiv f → (x y : A) → is-equiv (ap f {x} {y}) - is-emb-is-equiv {f} H x y = - is-equiv-is-invertible - ( λ p → - ( inv (is-retraction-map-inv-is-equiv H x)) ∙ - ( ( ap (map-inv-is-equiv H) p) ∙ - ( is-retraction-map-inv-is-equiv H y))) - ( λ p → - ( ap-concat f - ( inv (is-retraction-map-inv-is-equiv H x)) - ( ap (map-inv-is-equiv H) p ∙ is-retraction-map-inv-is-equiv H y)) ∙ - ( ( ap-binary - ( λ u v → u ∙ v) - ( ap-inv f (is-retraction-map-inv-is-equiv H x)) - ( ( ap-concat f - ( ap (map-inv-is-equiv H) p) - ( is-retraction-map-inv-is-equiv H y)) ∙ - ( ap-binary - ( λ u v → u ∙ v) - ( inv (ap-comp f (map-inv-is-equiv H) p)) - ( inv (coherence-map-inv-is-equiv H y))))) ∙ - ( inv - ( left-transpose-eq-concat - ( ap f (is-retraction-map-inv-is-equiv H x)) - ( p) - ( ( ap (f ∘ map-inv-is-equiv H) p) ∙ - ( is-section-map-inv-is-equiv H (f y))) - ( ( ap-binary - ( λ u v → u ∙ v) - ( inv (coherence-map-inv-is-equiv H x)) - ( inv (ap-id p))) ∙ - ( nat-htpy (is-section-map-inv-is-equiv H) p)))))) - ( λ where refl → left-inv (is-retraction-map-inv-is-equiv H x)) + is-emb-is-equiv H x y = + is-equiv-is-invertible' + ( is-invertible-ap-is-coherently-invertible + ( is-coherently-invertible-is-equiv H)) equiv-ap : (e : A ≃ B) (x y : A) → (x ＝ y) ≃ (map-equiv e x ＝ map-equiv e y) diff --git a/src/foundation-core/homotopies.lagda.md b/src/foundation-core/homotopies.lagda.md index fc6b8eb7de..9c53f25ce0 100644 --- a/src/foundation-core/homotopies.lagda.md +++ b/src/foundation-core/homotopies.lagda.md @@ -247,6 +247,20 @@ module _ ### Naturality of homotopies with respect to identifications +Given two maps `f g : A → B` and a homotopy `H : f ~ g`, then for every +identification `p : x ＝ y` in `A`, we have a +[commuting square of identifications](foundation-core.commuting-squares-of-identifications.md) + +```text + ap f p + f x -------> f y + | | + H x | | H y + ∨ ∨ + g x -------> g y. + ap g p +``` + ```agda nat-htpy : {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} (H : f ~ g) diff --git a/src/foundation-core/invertible-maps.lagda.md b/src/foundation-core/invertible-maps.lagda.md index 3392fd1794..0b7ba10a4c 100644 --- a/src/foundation-core/invertible-maps.lagda.md +++ b/src/foundation-core/invertible-maps.lagda.md @@ -7,12 +7,15 @@ module foundation-core.invertible-maps where
Imports ```agda +open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.universe-levels +open import foundation.whiskering-homotopies-composition open import foundation-core.cartesian-product-types open import foundation-core.function-types open import foundation-core.homotopies +open import foundation-core.identity-types open import foundation-core.retractions open import foundation-core.sections ``` @@ -39,11 +42,11 @@ module _ is-inverse f g = ((f ∘ g) ~ id) × ((g ∘ f) ~ id) is-section-is-inverse : - {f : A → B} {g : B → A} → is-inverse f g → (f ∘ g) ~ id + {f : A → B} {g : B → A} → is-inverse f g → f ∘ g ~ id is-section-is-inverse = pr1 is-retraction-is-inverse : - {f : A → B} {g : B → A} → is-inverse f g → (g ∘ f) ~ id + {f : A → B} {g : B → A} → is-inverse f g → g ∘ f ~ id is-retraction-is-inverse = pr2 ``` @@ -64,19 +67,19 @@ module _ is-inverse-map-inv-is-invertible : is-inverse f map-inv-is-invertible is-inverse-map-inv-is-invertible = pr2 g - is-retraction-is-invertible : (f ∘ map-inv-is-invertible) ~ id - is-retraction-is-invertible = pr1 is-inverse-map-inv-is-invertible + is-section-map-inv-is-invertible : f ∘ map-inv-is-invertible ~ id + is-section-map-inv-is-invertible = pr1 is-inverse-map-inv-is-invertible - is-section-is-invertible : (map-inv-is-invertible ∘ f) ~ id - is-section-is-invertible = pr2 is-inverse-map-inv-is-invertible + is-retraction-map-inv-is-invertible : map-inv-is-invertible ∘ f ~ id + is-retraction-map-inv-is-invertible = pr2 is-inverse-map-inv-is-invertible section-is-invertible : section f pr1 section-is-invertible = map-inv-is-invertible - pr2 section-is-invertible = is-retraction-is-invertible + pr2 section-is-invertible = is-section-map-inv-is-invertible retraction-is-invertible : retraction f pr1 retraction-is-invertible = map-inv-is-invertible - pr2 retraction-is-invertible = is-section-is-invertible + pr2 retraction-is-invertible = is-retraction-map-inv-is-invertible ``` ### The type of invertible maps @@ -100,37 +103,231 @@ module _ map-inv-invertible-map = map-inv-is-invertible ∘ is-invertible-map-invertible-map - is-section-map-invertible-map : + is-retraction-map-inv-invertible-map : (f : invertible-map A B) → - (map-inv-invertible-map f ∘ map-invertible-map f) ~ id - is-section-map-invertible-map = - is-section-is-invertible ∘ is-invertible-map-invertible-map + map-inv-invertible-map f ∘ map-invertible-map f ~ id + is-retraction-map-inv-invertible-map = + is-retraction-map-inv-is-invertible ∘ is-invertible-map-invertible-map - is-retraction-map-invertible-map : + is-section-map-inv-invertible-map : (f : invertible-map A B) → - (map-invertible-map f ∘ map-inv-invertible-map f) ~ id - is-retraction-map-invertible-map = - is-retraction-is-invertible ∘ is-invertible-map-invertible-map + map-invertible-map f ∘ map-inv-invertible-map f ~ id + is-section-map-inv-invertible-map = + is-section-map-inv-is-invertible ∘ is-invertible-map-invertible-map ``` ## Properties -### The invertible inverse of an invertible map +### The identity invertible map ```agda module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + {l1 : Level} {A : UU l1} + where + + is-inverse-id : is-inverse id (id {A = A}) + pr1 is-inverse-id = refl-htpy + pr2 is-inverse-id = refl-htpy + + is-invertible-id : is-invertible (id {A = A}) + pr1 is-invertible-id = id + pr2 is-invertible-id = is-inverse-id + + id-invertible-map : invertible-map A A + pr1 id-invertible-map = id + pr2 id-invertible-map = is-invertible-id +``` + +### The inverse of an invertible map + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-inverse-inv-is-inverse : + {f : A → B} {g : B → A} → is-inverse f g → is-inverse g f + pr1 (is-inverse-inv-is-inverse {f} {g} H) = + is-retraction-map-inv-is-invertible (g , H) + pr2 (is-inverse-inv-is-inverse {f} {g} H) = + is-section-map-inv-is-invertible (g , H) + + is-invertible-map-inv-is-invertible : + {f : A → B} (g : is-invertible f) → is-invertible (map-inv-is-invertible g) + pr1 (is-invertible-map-inv-is-invertible {f} g) = f + pr2 (is-invertible-map-inv-is-invertible {f} g) = + is-inverse-inv-is-inverse {f} (is-inverse-map-inv-is-invertible g) + + is-invertible-map-inv-invertible-map : + (f : invertible-map A B) → is-invertible (map-inv-invertible-map f) + is-invertible-map-inv-invertible-map f = + is-invertible-map-inv-is-invertible (is-invertible-map-invertible-map f) + + inv-invertible-map : invertible-map A B → invertible-map B A + pr1 (inv-invertible-map f) = map-inv-invertible-map f + pr2 (inv-invertible-map f) = is-invertible-map-inv-invertible-map f +``` + +### The inversion operation on invertible maps is a strict involution + +The inversion operation on invertible maps is a strict involution, where, by +strict involution, we mean that `inv-invertible-map (inv-invertible-map f) ≐ f` +syntactically. This can be observed by the fact that the type-checker accepts +`refl` as proof of this equation. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-involution-inv-invertible-map : + {f : invertible-map A B} → inv-invertible-map (inv-invertible-map f) ＝ f + is-involution-inv-invertible-map = refl +``` + +### Composition of invertible maps + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} + (g : B → C) (f : A → B) (G : is-invertible g) (F : is-invertible f) + where + + map-inv-is-invertible-comp : C → A + map-inv-is-invertible-comp = + map-inv-is-invertible F ∘ map-inv-is-invertible G + + is-section-map-inv-is-invertible-comp : + is-section (g ∘ f) map-inv-is-invertible-comp + is-section-map-inv-is-invertible-comp = + is-section-map-section-comp g f + ( section-is-invertible F) + ( section-is-invertible G) + + is-retraction-map-inv-is-invertible-comp : + is-retraction (g ∘ f) map-inv-is-invertible-comp + is-retraction-map-inv-is-invertible-comp = + is-retraction-map-retraction-comp g f + ( retraction-is-invertible G) + ( retraction-is-invertible F) + + is-invertible-comp : is-invertible (g ∘ f) + is-invertible-comp = + ( map-inv-is-invertible-comp , + is-section-map-inv-is-invertible-comp , + is-retraction-map-inv-is-invertible-comp) + +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} where - inv-is-inverse : {g : B → A} → is-inverse f g → is-inverse g f - pr1 (inv-is-inverse {g} H) = is-section-is-invertible (g , H) - pr2 (inv-is-inverse {g} H) = is-retraction-is-invertible (g , H) + is-invertible-map-comp-invertible-map : + (g : invertible-map B C) (f : invertible-map A B) → + is-invertible (map-invertible-map g ∘ map-invertible-map f) + is-invertible-map-comp-invertible-map g f = + is-invertible-comp + ( map-invertible-map g) + ( map-invertible-map f) + ( is-invertible-map-invertible-map g) + ( is-invertible-map-invertible-map f) + + comp-invertible-map : + invertible-map B C → invertible-map A B → invertible-map A C + pr1 (comp-invertible-map g f) = map-invertible-map g ∘ map-invertible-map f + pr2 (comp-invertible-map g f) = is-invertible-map-comp-invertible-map g f +``` + +### Invertible maps are closed under homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-section-map-inv-is-invertible-htpy : + {f f' : A → B} (H : f' ~ f) (F : is-invertible f) → + is-section f' (map-inv-is-invertible F) + is-section-map-inv-is-invertible-htpy H (g , S , R) = H ·r g ∙h S + + is-retraction-map-inv-is-invertible-htpy : + {f f' : A → B} (H : f' ~ f) (F : is-invertible f) → + is-retraction f' (map-inv-is-invertible F) + is-retraction-map-inv-is-invertible-htpy H (g , S , R) = g ·l H ∙h R + + is-invertible-htpy : + {f f' : A → B} → f' ~ f → is-invertible f → is-invertible f' + is-invertible-htpy H F = + ( map-inv-is-invertible F , + is-section-map-inv-is-invertible-htpy H F , + is-retraction-map-inv-is-invertible-htpy H F) + + is-invertible-inv-htpy : + {f f' : A → B} → f ~ f' → is-invertible f → is-invertible f' + is-invertible-inv-htpy H = is-invertible-htpy (inv-htpy H) + + htpy-map-inv-is-invertible : + {f g : A → B} (H : f ~ g) (F : is-invertible f) (G : is-invertible g) → + map-inv-is-invertible F ~ map-inv-is-invertible G + htpy-map-inv-is-invertible H F G = + ( ( inv-htpy (is-retraction-map-inv-is-invertible G)) ·r + ( map-inv-is-invertible F)) ∙h + ( ( map-inv-is-invertible G) ·l + ( ( inv-htpy H ·r map-inv-is-invertible F) ∙h + ( is-section-map-inv-is-invertible F))) +``` + +### Any section of an invertible map is homotopic to its inverse + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : invertible-map A B) + where + + htpy-map-inv-invertible-map-section : + (f : section (map-invertible-map e)) → + map-inv-invertible-map e ~ + map-section (map-invertible-map e) f + htpy-map-inv-invertible-map-section (f , H) = + ( map-inv-invertible-map e ·l inv-htpy H) ∙h + ( is-retraction-map-inv-invertible-map e ·r f) +``` + +### Any retraction of an invertible map is homotopic to its inverse + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : invertible-map A B) + where + + htpy-map-inv-invertible-map-retraction : + (f : retraction (map-invertible-map e)) → + map-inv-invertible-map e ~ + map-retraction (map-invertible-map e) f + htpy-map-inv-invertible-map-retraction (f , H) = + ( inv-htpy H ·r map-inv-invertible-map e) ∙h + ( f ·l is-section-map-inv-invertible-map e) +``` + +### Invertible maps are injective + +The construction of the converse map of the +[action on identifications](foundation.action-on-identifications-functions.md) +is a rerun of the proof that maps with +[retractions](foundation-core.retractions.md) are +[injective](foundation-core.injective-maps.md) (`is-injective-retraction`). We +repeat the proof to avoid cyclic dependencies. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + (H : is-invertible f) {x y : A} + where - inv-is-invertible : - (g : is-invertible f) → is-invertible (map-inv-is-invertible g) - pr1 (inv-is-invertible g) = f - pr2 (inv-is-invertible g) = - inv-is-inverse (is-inverse-map-inv-is-invertible g) + is-injective-is-invertible : f x ＝ f y → x ＝ y + is-injective-is-invertible p = + ( inv (is-retraction-map-inv-is-invertible H x)) ∙ + ( ( ap (map-inv-is-invertible H) p) ∙ + ( is-retraction-map-inv-is-invertible H y)) ``` ## See also diff --git a/src/foundation-core/whiskering-homotopies-concatenation.lagda.md b/src/foundation-core/whiskering-homotopies-concatenation.lagda.md new file mode 100644 index 0000000000..8eae6bc964 --- /dev/null +++ b/src/foundation-core/whiskering-homotopies-concatenation.lagda.md @@ -0,0 +1,133 @@ +# Whiskering homotopies with respect to concatenation + +```agda +module foundation-core.whiskering-homotopies-concatenation where +``` + +
Imports + +```agda +open import foundation.universe-levels +open import foundation.whiskering-operations + +open import foundation-core.homotopies +open import foundation-core.whiskering-identifications-concatenation +``` + +
+ +## Idea + +Consider a homotopy `H : f ~ g` and a homotopy `K : I ~ J` between two +homotopies `I J : g ~ f`. The +{{#concept "left whiskering" Disambiguation="homotopies with respect to concatenation" Agda=left-whisker-concat-htpy}} +of `H` and `K` is a homotopy `H ∙h I ~ H ∙h J`. In other words, left whiskering +of homotopies with respect to concatenation is a +[whiskering operation](foundation.whiskering-operations.md) + +```text + (H : f ~ g) {I J : g ~ h} → I ~ J → H ∙h I ~ H ∙h K. +``` + +Similarly, we introduce +{{#concept "right whiskering" Disambiguation="homotopies with respect to concatenation" Agda=right-whisker-concat-htpy}} +to be an operation + +```text + {H I : f ~ g} → H ~ I → (J : g ~ h) → H ∙h J ~ I ∙h J. +``` + +## Definitions + +### Left whiskering of homotopies with respect to concatenation + +Left whiskering of homotopies with respect to concatenation is an operation + +```text + (H : f ~ g) {I J : g ~ h} → I ~ J → H ∙h I ~ H ∙h J. +``` + +We implement the left whiskering operation of homotopies with respect to +concatenation as an instance of a general left whiskering operation. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + left-whisker-concat-htpy : + left-whiskering-operation ((x : A) → B x) (_~_) (_∙h_) (_~_) + left-whisker-concat-htpy H K x = left-whisker-concat (H x) (K x) + + left-unwhisker-concat-htpy : + {f g h : (x : A) → B x} (H : f ~ g) {I J : g ~ h} → H ∙h I ~ H ∙h J → I ~ J + left-unwhisker-concat-htpy H K x = left-unwhisker-concat (H x) (K x) +``` + +### Right whiskering of homotopies with respect to concatenation + +Right whiskering of homotopies with respect to concatenation is an operation + +```text + {H I : f ~ g} → H ~ I → (J : g ~ h) → H ∙h J ~ I ∙h J. +``` + +We implement the right whiskering operation of homotopies with respect to +concatenation as an instance of a general right whiskering operation. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + right-whisker-concat-htpy : + right-whiskering-operation ((x : A) → B x) (_~_) (_∙h_) (_~_) + right-whisker-concat-htpy K J x = right-whisker-concat (K x) (J x) + + right-unwhisker-concat-htpy : + {f g h : (x : A) → B x} {H I : f ~ g} (J : g ~ h) → H ∙h J ~ I ∙h J → H ~ I + right-unwhisker-concat-htpy H K x = right-unwhisker-concat (H x) (K x) +``` + +## Properties + +### The unit and absorption laws for left whiskering of homotopies with respect to concatenation + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + left-unit-law-left-whisker-concat-htpy : + {f g : (x : A) → B x} {I J : f ~ g} (K : I ~ J) → + left-whisker-concat-htpy refl-htpy K ~ K + left-unit-law-left-whisker-concat-htpy K x = + left-unit-law-left-whisker-concat (K x) + + right-absorption-law-left-whisker-concat-htpy : + {f g h : (x : A) → B x} (H : f ~ g) {I : g ~ h} → + left-whisker-concat-htpy H (refl-htpy' I) ~ refl-htpy + right-absorption-law-left-whisker-concat-htpy H x = + right-absorption-law-left-whisker-concat (H x) _ +``` + +### The unit and absorption laws for right whiskering of homotopies with respect to concatenation + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + left-absorption-law-right-whisker-concat-htpy : + {f g h : (x : A) → B x} {H : f ~ g} (J : g ~ h) → + right-whisker-concat-htpy (refl-htpy' H) J ~ refl-htpy + left-absorption-law-right-whisker-concat-htpy J x = + left-absorption-law-right-whisker-concat _ (J x) + + right-unit-law-right-whisker-concat-htpy : + {f g : (x : A) → B x} {I J : f ~ g} (K : I ~ J) → + right-unit-htpy ∙h K ~ + right-whisker-concat-htpy K refl-htpy ∙h right-unit-htpy + right-unit-law-right-whisker-concat-htpy K x = + right-unit-law-right-whisker-concat (K x) +``` diff --git a/src/foundation-core/whiskering-identifications-concatenation.lagda.md b/src/foundation-core/whiskering-identifications-concatenation.lagda.md new file mode 100644 index 0000000000..9710a2cb0e --- /dev/null +++ b/src/foundation-core/whiskering-identifications-concatenation.lagda.md @@ -0,0 +1,333 @@ +# Whiskering identifications with respect to concatenation + +```agda +module foundation-core.whiskering-identifications-concatenation where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.universe-levels +open import foundation.whiskering-operations + +open import foundation-core.homotopies +open import foundation-core.identity-types +``` + +
+ +## Idea + +Consider two [identifications](foundation-core.identity-types.md) `p q : x ＝ y` +in a type `A`. The whiskering operations are operations that take +identifications `p ＝ q` to identifications `r ∙ p ＝ r ∙ q` or to +identifications `p ∙ r ＝ q ∙ r`. + +The +{{#concept "left whiskering" Disambiguation="identifications" Agda=left-whisker-concat}} +operation takes an identification `r : z ＝ x` and an identification `p ＝ q` to +an identification `r ∙ p ＝ r ∙ q`. Similarly, the +{{#concept "right whiskering" Disambiguation="identifications" Agda=right-whisker-concat}} +operation takes an identification `r : y ＝ z` and an identification `p ＝ q` to +an identification `p ∙ r ＝ q ∙ r`. + +The whiskering operations can be defined by the +[acion on identifications](foundation.action-on-identifications-functions.md) of +concatenation. Since concatenation on either side is an +[equivalence](foundation-core.equivalences.md), it follows that the whiskering +operations are equivalences. + +## Definitions + +### Left whiskering of identifications + +Left whiskering of identifications with respect to concatenation is an operation + +```text + (p : x ＝ y) {q r : y ＝ z} → q ＝ r → p ∙ q ＝ p ∙ r +``` + +on any type. + +```agda +module _ + {l : Level} {A : UU l} + where + + left-whisker-concat : left-whiskering-operation A (_＝_) (_∙_) (_＝_) + left-whisker-concat p β = ap (p ∙_) β + + left-unwhisker-concat : + {x y z : A} (p : x ＝ y) {q r : y ＝ z} → p ∙ q ＝ p ∙ r → q ＝ r + left-unwhisker-concat = is-injective-concat +``` + +### Right whiskering of identifications + +Right whiskering of identifications with respect to concatenation is an +operation + +```text + {p q : x ＝ y} → p ＝ q → (r : y ＝ z) → p ∙ r ＝ q ∙ r +``` + +on any type. + +```agda +module _ + {l : Level} {A : UU l} + where + + right-whisker-concat : right-whiskering-operation A (_＝_) (_∙_) (_＝_) + right-whisker-concat α q = ap (_∙ q) α + + right-unwhisker-concat : + {x y z : A} {p q : x ＝ y} (r : y ＝ z) → p ∙ r ＝ q ∙ r → p ＝ q + right-unwhisker-concat r = is-injective-concat' r +``` + +### Double whiskering of identifications + +```agda +module _ + {l : Level} {A : UU l} + {a b c d : A} (p : a ＝ b) {r s : b ＝ c} (t : r ＝ s) (q : c ＝ d) + where + + double-whisker-concat : (p ∙ r) ∙ q ＝ (p ∙ s) ∙ q + double-whisker-concat = right-whisker-concat (left-whisker-concat p t) q + + double-whisker-concat' : p ∙ (r ∙ q) ＝ p ∙ (s ∙ q) + double-whisker-concat' = left-whisker-concat p (right-whisker-concat t q) +``` + +## Properties + +### The unit and absorption laws for left whiskering of identifications + +```agda +module _ + {l : Level} {A : UU l} + where + + left-unit-law-left-whisker-concat : + {x y : A} {p p' : x ＝ y} (α : p ＝ p') → + left-whisker-concat refl α ＝ α + left-unit-law-left-whisker-concat refl = refl + + right-absorption-law-left-whisker-concat : + {x y z : A} (p : x ＝ y) (q : y ＝ z) → + left-whisker-concat p (refl {x = q}) ＝ refl + right-absorption-law-left-whisker-concat p q = refl +``` + +### The unit and absorption laws for right whiskering of identifications + +The right unit law for right whiskering of identifications with respect to +concatenation asserts that the square of identifications + +```text + right-whisker-concat α refl + p ∙ refl -----------------------------> p' ∙ refl + | | + right-unit | | + ∨ ∨ + p -------------------------------------> p' +``` + +commutes for any `α : p ＝ p'`. Note that this law is slightly more complicated, +since concatenating with `refl` on the right does not compute to the identity +function. + +```agda +module _ + {l : Level} {A : UU l} + where + + right-unit-law-right-whisker-concat : + {x y : A} {p p' : x ＝ y} (α : p ＝ p') → + right-unit ∙ α ＝ right-whisker-concat α refl ∙ right-unit + right-unit-law-right-whisker-concat {p = refl} refl = refl + + left-absorption-law-right-whisker-concat : + {x y z : A} (p : x ＝ y) (q : y ＝ z) → + right-whisker-concat (refl {x = p}) q ＝ refl + left-absorption-law-right-whisker-concat p q = refl +``` + +### Commutativity of left and right whiskering of identifications + +Consider four identifications `p p' : x ＝ y` and `q q' : y ＝ z` in a type `A`. +Then the square of identifications + +```text + right-whisker α q + p ∙ q ---------------------> p' ∙ q + | | + left-whisker p β | | left-whisker p' β + ∨ ∨ + p ∙ q' --------------------> p' ∙ q' + right-whisker α q' +``` + +commutes. There are at least two natural ways in which this square is seen to +commute: + +1. The square commutes by naturality of the homotopy + `α ↦ left-whisker-concat α β`. +2. The transposed square commutes by the naturality of the homotopy + `β ↦ right-whisker-concat α β`. + +These two ways in which the square commutes are inverse to each other. + +**Note.** The following statements could have been formalized using +[commuting squares of identifications](foundation.commuting-squares-of-identifications.md). +However, in order to avoid cyclic module dependencies in the library we avoid +doing so. + +```agda +module _ + {l : Level} {A : UU l} {x y z : A} + where + + commutative-left-whisker-right-whisker-concat : + {q q' : y ＝ z} (β : q ＝ q') {p p' : x ＝ y} (α : p ＝ p') → + left-whisker-concat p β ∙ right-whisker-concat α q' ＝ + right-whisker-concat α q ∙ left-whisker-concat p' β + commutative-left-whisker-right-whisker-concat β = + nat-htpy (λ α → left-whisker-concat α β) + + commutative-right-whisker-left-whisker-concat : + {p p' : x ＝ y} (α : p ＝ p') {q q' : y ＝ z} (β : q ＝ q') → + right-whisker-concat α q ∙ left-whisker-concat p' β ＝ + left-whisker-concat p β ∙ right-whisker-concat α q' + commutative-right-whisker-left-whisker-concat α = + nat-htpy (right-whisker-concat α) + + compute-inv-commutative-left-whisker-right-whisker-concat : + {q q' : y ＝ z} (β : q ＝ q') {p p' : x ＝ y} (α : p ＝ p') → + inv (commutative-right-whisker-left-whisker-concat α β) ＝ + commutative-left-whisker-right-whisker-concat β α + compute-inv-commutative-left-whisker-right-whisker-concat refl refl = + refl +``` + +### Swapping the order of left and right whiskering of identifications + +Consider a diagram of identifications + +```text + r + p -----> q + a -----> b -----> c -----> + s +``` + +with `t : r ＝ s`. Then the square of identifications + +```text + assoc p r q + (p ∙ r) ∙ q -------------> p ∙ (r ∙ q) + | | + double-whisker p t q | | double-whisker' p t q + ∨ ∨ + (p ∙ s) ∙ q -------------> p ∙ (s ∙ q) + assoc p s q +``` + +commutes. + +```agda +module _ + {l1 : Level} {A : UU l1} + where + + swap-double-whisker-concat : + {a b c d : A} (p : a ＝ b) {r s : b ＝ c} (t : r ＝ s) (q : c ＝ d) → + double-whisker-concat p t q ∙ assoc p s q ＝ + assoc p r q ∙ double-whisker-concat' p t q + swap-double-whisker-concat refl refl refl = refl +``` + +### The action on identifications of concatenating by `refl` on the right + +Consider an identification `r : p ＝ q` between two identifications +`p q : x ＝ y` in a type `A`. Then the square of identifications + +```text + right-whisker r refl + p ∙ refl ----------------------> q ∙ refl + | | + right-unit | | right-unit + ∨ ∨ + p -----------------------------> q + r +``` + +commutes. + +```agda +module _ + {l : Level} {A : UU l} {x y : A} {p q : x ＝ y} + where + + compute-refl-right-whisker-concat : + (r : p ＝ q) → + right-unit ∙ r ＝ right-whisker-concat r refl ∙ right-unit + compute-refl-right-whisker-concat refl = right-unit +``` + +### Left whiskering of identifications distributes over concatenation + +```agda +module _ + {l : Level} {A : UU l} + where + + distributive-left-whisker-concat-concat : + {a b c : A} (p : a ＝ b) {q r s : b ＝ c} (α : q ＝ r) (β : r ＝ s) → + left-whisker-concat p (α ∙ β) ＝ + left-whisker-concat p α ∙ left-whisker-concat p β + distributive-left-whisker-concat-concat p refl β = refl +``` + +### Right whiskering of identifications distributes over concatenation + +```agda +module _ + {l : Level} {A : UU l} + where + + distributive-right-whisker-concat-concat : + {a b c : A} {p q r : a ＝ b} (α : p ＝ q) (β : q ＝ r) (s : b ＝ c) → + right-whisker-concat (α ∙ β) s ＝ + right-whisker-concat α s ∙ right-whisker-concat β s + distributive-right-whisker-concat-concat refl β s = refl +``` + +### Left whiskering of identifications commutes with inverses of identifications + +```agda +module _ + {l : Level} {A : UU l} + where + + compute-inv-left-whisker-concat : + {a b c : A} (p : a ＝ b) {q r : b ＝ c} (s : q ＝ r) → + left-whisker-concat p (inv s) ＝ inv (left-whisker-concat p s) + compute-inv-left-whisker-concat p s = ap-inv (concat p _) s +``` + +### Right whiskering of identifications commutes with inverses of identifications + +```agda +module _ + {l : Level} {A : UU l} + where + + compute-inv-right-whisker-concat : + {a b c : A} {p q : a ＝ b} (s : p ＝ q) (r : b ＝ c) → + right-whisker-concat (inv s) r ＝ inv (right-whisker-concat s r) + compute-inv-right-whisker-concat s r = ap-inv (concat' _ r) s +``` diff --git a/src/foundation/action-on-higher-identifications-functions.lagda.md b/src/foundation/action-on-higher-identifications-functions.lagda.md index 1553a81b60..b09a5cf131 100644 --- a/src/foundation/action-on-higher-identifications-functions.lagda.md +++ b/src/foundation/action-on-higher-identifications-functions.lagda.md @@ -8,10 +8,10 @@ module foundation.action-on-higher-identifications-functions where ```agda open import foundation.action-on-identifications-functions -open import foundation.commuting-squares-of-identifications open import foundation.path-algebra open import foundation.universe-levels +open import foundation-core.commuting-squares-of-identifications open import foundation-core.constant-maps open import foundation-core.function-types open import foundation-core.homotopies diff --git a/src/foundation/action-on-homotopies-functions.lagda.md b/src/foundation/action-on-homotopies-functions.lagda.md index cd4c2023db..0bb5100c4c 100644 --- a/src/foundation/action-on-homotopies-functions.lagda.md +++ b/src/foundation/action-on-homotopies-functions.lagda.md @@ -11,13 +11,13 @@ open import foundation.action-on-higher-identifications-functions open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.function-extensionality -open import foundation.homotopies open import foundation.homotopy-induction open import foundation.universe-levels open import foundation-core.constant-maps open import foundation-core.contractible-types open import foundation-core.function-types +open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/automorphisms.lagda.md b/src/foundation/automorphisms.lagda.md index bc5d12d351..183752d129 100644 --- a/src/foundation/automorphisms.lagda.md +++ b/src/foundation/automorphisms.lagda.md @@ -8,10 +8,10 @@ module foundation.automorphisms where ```agda open import foundation.dependent-pair-types -open import foundation.sets open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.sets open import structured-types.pointed-types ``` diff --git a/src/foundation/coherently-invertible-maps.lagda.md b/src/foundation/coherently-invertible-maps.lagda.md index 9bc14bb139..d43a5f5702 100644 --- a/src/foundation/coherently-invertible-maps.lagda.md +++ b/src/foundation/coherently-invertible-maps.lagda.md @@ -15,15 +15,16 @@ open import foundation.equivalences open import foundation.identity-types open import foundation.type-arithmetic-dependent-pair-types open import foundation.universe-levels -open import foundation.whiskering-homotopies-composition +open import foundation.whiskering-higher-homotopies-composition +open import foundation-core.commuting-squares-of-homotopies open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.fibers-of-maps -open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositions +open import foundation-core.retractions open import foundation-core.sections open import foundation-core.type-theoretic-principle-of-choice ``` @@ -32,68 +33,100 @@ open import foundation-core.type-theoretic-principle-of-choice ## Properties +### Coherently invertible maps have a contractible type of sections + +**Proof:** Since coherently invertible maps are +[contractible maps](foundation.contractible-maps.md), and products of +[contractible types](foundation-core.contractible-types.md) are contractible, it +follows that the type + +```text + (b : B) → fiber f b +``` + +is contractible, for any coherently invertible map `f`. However, by the +[type theoretic principle of choice](foundation.type-theoretic-principle-of-choice.md) +it follows that this type is equivalent to the type + +```text + Σ (B → A) (λ g → (b : B) → f (g b) ＝ b), +``` + +which is the type of [sections](foundation.sections.md) of `f`. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + abstract + is-contr-section-is-coherently-invertible : + {f : A → B} → is-coherently-invertible f → is-contr (section f) + is-contr-section-is-coherently-invertible {f} F = + is-contr-equiv' + ( (b : B) → fiber f b) + ( distributive-Π-Σ) + ( is-contr-Π (is-contr-map-is-coherently-invertible F)) +``` + ### Being coherently invertible is a property ```agda -abstract - is-prop-is-coherently-invertible : - {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → - is-prop (is-coherently-invertible f) - is-prop-is-coherently-invertible {A = A} {B} f = - is-prop-is-proof-irrelevant - ( λ H → - is-contr-equiv' - ( Σ ( section f) - ( λ s → - Σ ( ( map-section f s ∘ f) ~ id) - ( λ H → - ( right-whisker-comp (is-section-map-section f s) f) ~ - ( left-whisker-comp f H)))) - ( associative-Σ - ( B → A) - ( λ g → (f ∘ g) ~ id) - ( λ s → - Σ ( ( map-section f s ∘ f) ~ id) - ( λ H → - ( right-whisker-comp (is-section-map-section f s) f) ~ - ( left-whisker-comp f H)))) - ( is-contr-Σ - ( is-contr-section-is-equiv (is-equiv-is-coherently-invertible H)) - ( section-is-coherently-invertible H) - ( is-contr-equiv' - ( (x : A) → - Σ ( map-inv-is-coherently-invertible H (f x) ＝ x) - ( λ p → - is-retraction-is-coherently-invertible H (f x) ＝ ap f p)) - ( distributive-Π-Σ) - ( is-contr-Π - ( λ x → - is-contr-equiv' - ( fiber - ( ap f) - ( is-retraction-is-coherently-invertible H (f x))) - ( equiv-tot - ( λ p → - equiv-inv - ( ap f p) - ( is-retraction-is-coherently-invertible H (f x)))) - ( is-contr-map-is-equiv - ( is-emb-is-equiv - ( is-equiv-is-coherently-invertible H) - ( map-inv-is-coherently-invertible H (f x)) x) - ( is-retraction-is-coherently-invertible H (f x)))))))) - -abstract - is-equiv-is-coherently-invertible-is-equiv : - {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → - is-equiv (is-coherently-invertible-is-equiv {f = f}) - is-equiv-is-coherently-invertible-is-equiv f = - is-equiv-is-prop - ( is-property-is-equiv f) - ( is-prop-is-coherently-invertible f) - ( is-equiv-is-coherently-invertible) +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) + where + + abstract + is-proof-irrelevant-is-coherently-invertible : + is-proof-irrelevant (is-coherently-invertible f) + is-proof-irrelevant-is-coherently-invertible H = + is-contr-equiv' + ( _) + ( associative-Σ _ _ _) + ( is-contr-Σ + ( is-contr-section-is-coherently-invertible H) + ( section-is-coherently-invertible H) + ( is-contr-equiv' + ( _) + ( distributive-Π-Σ) + ( is-contr-Π + ( λ x → + is-contr-equiv' + ( _) + ( equiv-tot + ( λ p → + equiv-inv + ( ap f p) + ( is-section-map-inv-is-coherently-invertible H (f x)))) + ( is-contr-map-is-coherently-invertible + ( is-coherently-invertible-ap-is-coherently-invertible H) + ( is-section-map-inv-is-coherently-invertible H (f x))))))) + + abstract + is-prop-is-coherently-invertible : is-prop (is-coherently-invertible f) + is-prop-is-coherently-invertible = + is-prop-is-proof-irrelevant is-proof-irrelevant-is-coherently-invertible + + abstract + is-equiv-is-coherently-invertible-is-equiv : + is-equiv (is-coherently-invertible-is-equiv {f = f}) + is-equiv-is-coherently-invertible-is-equiv = + is-equiv-is-prop + ( is-property-is-equiv f) + ( is-prop-is-coherently-invertible) + ( is-equiv-is-coherently-invertible) ``` +### Being transpose coherently invertible is a property + +This remains to be formalized. + +## References + +1. Univalent Foundations Project, _Homotopy Type Theory – Univalent Foundations + of Mathematics_ (2013) ([website](https://homotopytypetheory.org/book/), + [arXiv:1308.0729](https://arxiv.org/abs/1308.0729)) + ## See also - For the notion of biinvertible maps see diff --git a/src/foundation/commuting-cubes-of-maps.lagda.md b/src/foundation/commuting-cubes-of-maps.lagda.md index 47d6b48201..13e4c00e42 100644 --- a/src/foundation/commuting-cubes-of-maps.lagda.md +++ b/src/foundation/commuting-cubes-of-maps.lagda.md @@ -16,11 +16,11 @@ open import foundation.function-extensionality open import foundation.homotopies open import foundation.universe-levels open import foundation.whiskering-homotopies-composition -open import foundation.whiskering-identifications-concatenation open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.precomposition-functions +open import foundation-core.whiskering-identifications-concatenation ```
diff --git a/src/foundation/commuting-prisms-of-maps.lagda.md b/src/foundation/commuting-prisms-of-maps.lagda.md index de583eb4b9..0c374973c9 100644 --- a/src/foundation/commuting-prisms-of-maps.lagda.md +++ b/src/foundation/commuting-prisms-of-maps.lagda.md @@ -10,7 +10,6 @@ open import foundation-core.commuting-prisms-of-maps public ```agda open import foundation.action-on-identifications-functions -open import foundation.commuting-squares-of-homotopies open import foundation.commuting-squares-of-maps open import foundation.commuting-triangles-of-maps open import foundation.composition-algebra @@ -22,6 +21,7 @@ open import foundation.precomposition-functions open import foundation.universe-levels open import foundation.whiskering-homotopies-composition +open import foundation-core.commuting-squares-of-homotopies open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types @@ -96,13 +96,22 @@ module _ ( right-whisker-concat-coherence-square-homotopies ( front-bottom ·r hA) ( bottom ·r hA' ·r hA) + ( hC' ·l mid ·r hA) + ( ( pasting-horizontal-coherence-square-maps + h' g' hA' hB' hC' h'' g'' left-bottom right-bottom) ·r + ( hA)) + ( prism-bottom ·r hA) ( hC' ·l ((g' ·l left-top) ∙h (right-top ·r h))) - ( prism-bottom ·r hA)) ∙h + ) ∙h ( ap-concat-htpy ( front-bottom ·r hA) ( ( map-coherence-square-homotopies hC' ( front-top) ( mid ·r hA) + (hC ·l top) + ( pasting-horizontal-coherence-square-maps h g hA hB hC h' g' + ( left-top) + ( right-top)) ( prism-top)) ∙h ( ap-concat-htpy ( hC' ·l front-top) diff --git a/src/foundation/commuting-squares-of-homotopies.lagda.md b/src/foundation/commuting-squares-of-homotopies.lagda.md index 5ae628881b..fa4657be1f 100644 --- a/src/foundation/commuting-squares-of-homotopies.lagda.md +++ b/src/foundation/commuting-squares-of-homotopies.lagda.md @@ -10,282 +10,534 @@ open import foundation-core.commuting-squares-of-homotopies public ```agda open import foundation.commuting-squares-of-identifications -open import foundation.functoriality-dependent-function-types +open import foundation.dependent-pair-types open import foundation.universe-levels -open import foundation.whiskering-homotopies-composition open import foundation-core.equivalences +open import foundation-core.functoriality-dependent-function-types open import foundation-core.homotopies +open import foundation-core.identity-types ```
## Idea -A square of [homotopies](foundation-core.homotopies.md) +A square of [homotopies](foundation-core.identity-types.md) ```text - top - f ------> g - | | - left | | right - v v - h ------> i - bottom + top + f -------> g + | | + left | | right + ∨ ∨ + h -------> i + bottom ``` -is said to be a {{#concept "commuting square" Disambiguation="homotopies"}} of -homotopies if there is a homotopy `left ∙h bottom ~ top ∙h right `. Such a -homotopy is called a +is said to be a +{{#concept "commuting square" Disambiguation="homotopies" Agda=coherence-square-homotopies}} +if there is a homotopy `left ∙h bottom ~ top ∙h right`. Such a homotopy is +called a {{#concept "coherence" Disambiguation="commuting square of homotopies" Agda=coherence-square-homotopies}} of the square. -### Right whiskering a commuting square of homotopies with respect to concatenation of homotopies +### Concatenating homotopies of edges and coherences of commuting squares of homotopies -Consider a -[commuting square of homotopies](foundation.commuting-squares-of-homotopies.md) +Consider a commuting square of homotopies and a homotopy of one of the four +sides with another homotopy, as for example in the diagram below: ```text - top - f ------> g - | | - left | | right - v v - h ------> i - bottom + top + a ---------> b + | | | + left | right |~| right' + ∨ ∨ ∨ + c ---------> d. + bottom ``` -and consider a homotopy `H : i ~ j`. Then there is an equivalence of commuting -squares of homotopies +Then any homotopy witnessing that the square commutes can be concatenated with +the homotopy on the side, to obtain a new commuting square of homotopies. + +#### Concatenating homotopies of the top edge with a coherence of a commuting square of homotopies + +Consider a commuting diagram of homotopies ```text - top top - f ------> g f -------------> g - | | | | - left | | right ≃ left | | right ∙h H - ∨ ∨ ∨ ∨ - h ------> i h -------------> j - bottom bottom ∙h H + top' + -------> + f -------> g + | top | + left | | right + ∨ ∨ + h -------> i. + bottom ``` -This is the -{{#concept "right whiskering" Disambiguation="commuting squares of homotopies with respect to concatenation" Agda=right-whisker-concat-coherence-square-homotopies}} -operation of commuting squares of homotopies with respect to concatenation. +with a homotopy `top ~ top'`. Then we get an equivalence + +```text + top top' + f -------> g f -------> g + | | | | + left | | right ≃ left | | right + ∨ ∨ ∨ ∨ + h -------> i h -------> i. + bottom bottom +``` ```agda module _ - { l1 l2 : Level} {A : UU l1} {B : UU l2} - { f g g' h k : A → B} - ( H : f ~ g) (H' : f ~ g') {K : g ~ h} {K' : g' ~ h} (L : h ~ k) + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + {top' : f ~ g} (s : top ~ top') where - equiv-right-whisker-concat-coherence-square-homotopies : - coherence-square-homotopies H H' K K' ≃ - coherence-square-homotopies H H' (K ∙h L) (K' ∙h L) - equiv-right-whisker-concat-coherence-square-homotopies = - equiv-Π-equiv-family - ( λ a → - equiv-right-whisker-concat-coherence-square-identifications - ( H a) - ( H' a) - ( K a) - ( K' a) - ( L a)) - - right-whisker-concat-coherence-square-homotopies : - coherence-square-homotopies H H' K K' → - coherence-square-homotopies H H' (K ∙h L) (K' ∙h L) - right-whisker-concat-coherence-square-homotopies = - map-equiv equiv-right-whisker-concat-coherence-square-homotopies - - right-unwhisker-concat-htpy-coherence-square-homotopies : - coherence-square-homotopies H H' (K ∙h L) (K' ∙h L) → - coherence-square-homotopies H H' K K' - right-unwhisker-concat-htpy-coherence-square-homotopies = - map-inv-equiv equiv-right-whisker-concat-coherence-square-homotopies + abstract + is-equiv-concat-top-homotopy-coherence-square-homotopies : + is-equiv + ( concat-top-homotopy-coherence-square-homotopies + top left right bottom s) + is-equiv-concat-top-homotopy-coherence-square-homotopies = + is-equiv-map-Π-is-fiberwise-equiv + ( λ x → + is-equiv-concat-top-identification-coherence-square-identifications + ( top x) (left x) (right x) (bottom x) (s x)) + + equiv-concat-top-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies top' left right bottom + pr1 equiv-concat-top-homotopy-coherence-square-homotopies = + concat-top-homotopy-coherence-square-homotopies top left right bottom s + pr2 equiv-concat-top-homotopy-coherence-square-homotopies = + is-equiv-concat-top-homotopy-coherence-square-homotopies ``` -### Left whiskering a commuting square of homotopies with respect to concatenation of homotopies +#### Concatenating homotopies of the left edge with a coherence of a commuting square of homotopies + +Consider a commuting diagram of homotopies + +```text + top + f -------> g + | | | + left' | | left | right + ∨ ∨ ∨ + h -------> i. + bottom +``` -Consider a -[commuting square of homotopies](foundation.commuting-squares-of-homotopies.md) +with a homotopy `left ~ left'`. Then we get an equivalence ```text - top - f ------> g - | | - left | | right - v v - h ------> i - bottom + top top + f -------> g f -------> g + | | | | + left | | right ≃ left' | | right + ∨ ∨ ∨ ∨ + h -------> i h -------> i. + bottom bottom ``` -and consider a homotopy `H : e ~ f`. Then there is an equivalence of commuting -squares of homotopies +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + {left' : f ~ h} (s : left ~ left') + where + + abstract + is-equiv-concat-left-homotopy-coherence-square-homotopies : + is-equiv + ( concat-left-homotopy-coherence-square-homotopies + top left right bottom s) + is-equiv-concat-left-homotopy-coherence-square-homotopies = + is-equiv-map-Π-is-fiberwise-equiv + ( λ x → + is-equiv-concat-left-identification-coherence-square-identifications + ( top x) (left x) (right x) (bottom x) (s x)) + + equiv-concat-left-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies top left' right bottom + pr1 equiv-concat-left-homotopy-coherence-square-homotopies = + concat-left-homotopy-coherence-square-homotopies top left right bottom s + pr2 equiv-concat-left-homotopy-coherence-square-homotopies = + is-equiv-concat-left-homotopy-coherence-square-homotopies +``` + +#### Concatenating homotopies of the right edge with a coherence of a commuting square of homotopies + +Consider a commuting diagram of homotopies ```text - top H ∙h top - f ------> g e ----------> g - | | | | - left | | right ≃ H ∙h left | | right - ∨ ∨ ∨ ∨ - h ------> i h ----------> i - bottom bottom + top + f -------> g + | | | + left | right | | right' + ∨ ∨ ∨ + h -------> i. + bottom ``` -This is the -{{#concept "left whiskering" Disambiguation="commuting squares of homotopies with respect to concatenation" Agda=left-whisker-concat-coherence-square-homotopies}} -operation of commuting squares of homotopies with respect to concatenation. +with a homotopy `right ~ right'`. Then we get an equivalence + +```text + top top + f -------> g f -------> g + | | | | + left | | right ≃ left | | right' + ∨ ∨ ∨ ∨ + h -------> i h -------> i. + bottom bottom +``` ```agda module _ - { l1 l2 : Level} {A : UU l1} {B : UU l2} - { f g g' h k : A → B} - ( L : k ~ f) {H : f ~ g} {H' : f ~ g'} {K : g ~ h} {K' : g' ~ h} + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + {right' : g ~ i} (s : right ~ right') where - equiv-left-whisker-concat-coherence-square-homotopies : - ( coherence-square-homotopies H H' K K') ≃ - ( coherence-square-homotopies (L ∙h H) (L ∙h H') K K') - equiv-left-whisker-concat-coherence-square-homotopies = - equiv-Π-equiv-family - ( λ a → - equiv-left-whisker-concat-coherence-square-identifications - ( L a) - ( H a) - ( H' a) - ( K a) - ( K' a)) - - left-whisker-concat-coherence-square-homotopies : - coherence-square-homotopies H H' K K' → - coherence-square-homotopies (L ∙h H) (L ∙h H') K K' - left-whisker-concat-coherence-square-homotopies = - map-equiv equiv-left-whisker-concat-coherence-square-homotopies - - left-unwhisker-concat-htpy-coherence-square-homotopies : - coherence-square-homotopies (L ∙h H) (L ∙h H') K K' → - coherence-square-homotopies H H' K K' - left-unwhisker-concat-htpy-coherence-square-homotopies = - map-inv-equiv equiv-left-whisker-concat-coherence-square-homotopies + abstract + is-equiv-concat-right-homotopy-coherence-square-homotopies : + is-equiv + ( concat-right-homotopy-coherence-square-homotopies + top left right bottom s) + is-equiv-concat-right-homotopy-coherence-square-homotopies = + is-equiv-map-Π-is-fiberwise-equiv + ( λ x → + is-equiv-concat-right-identification-coherence-square-identifications + ( top x) (left x) (right x) (bottom x) (s x)) + + equiv-concat-right-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies top left right' bottom + pr1 equiv-concat-right-homotopy-coherence-square-homotopies = + concat-right-homotopy-coherence-square-homotopies top left right bottom s + pr2 equiv-concat-right-homotopy-coherence-square-homotopies = + is-equiv-concat-right-homotopy-coherence-square-homotopies +``` + +#### Concatenating homotopies of the bottom edge with a coherence of a commuting square of homotopies + +Consider a commuting diagram of homotopies + +```text + top + f -------> g + | | + left | | right + ∨ bottom ∨ + h -------> i. + -------> + bottom' +``` + +with a homotopy `bottom ~ bottom'`. Then we get an equivalence + +```text + top top + f -------> g f -------> g + | | | | + left | | right ≃ left | | right + ∨ ∨ ∨ ∨ + h -------> i h -------> i. + bottom bottom' +``` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + {bottom' : h ~ i} (s : bottom ~ bottom') + where + + is-equiv-concat-bottom-homotopy-coherence-square-homotopies : + is-equiv + ( concat-bottom-homotopy-coherence-square-homotopies + top left right bottom s) + is-equiv-concat-bottom-homotopy-coherence-square-homotopies = + is-equiv-map-Π-is-fiberwise-equiv + ( λ x → + is-equiv-concat-bottom-identification-coherence-square-identifications + ( top x) (left x) (right x) (bottom x) (s x)) + + equiv-concat-bottom-homotopy-coherence-square-homotopies : + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies top left right bottom' + pr1 equiv-concat-bottom-homotopy-coherence-square-homotopies = + concat-bottom-homotopy-coherence-square-homotopies top left right bottom s + pr2 equiv-concat-bottom-homotopy-coherence-square-homotopies = + is-equiv-concat-bottom-homotopy-coherence-square-homotopies +``` + +### Whiskering and splicing coherences of commuting squares of homotopies + +Given a commuting square of homotopies + +```text + top + f -------> g + | | + left | | right + ∨ ∨ + h -------> i, + bottom ``` -### Left whiskering a commuting square of homotopies with respect to concatenation of homotopies +we may consider four ways of attaching new homotopies to it: + +1. Prepending `p : u ~ f` to the left gives us a commuting square + + ```text + p ∙h top + u -------> g + | | + p ∙h left | | right + ∨ ∨ + h -------> i. + bottom + ``` + + More precisely, we have an equivalence -Consider a -[commuting square of homotopies](foundation.commuting-squares-of-homotopies.md) + ```text + (left ∙h bottom ~ top ∙h right) ≃ ((p ∙h left) ∙h bottom ~ (p ∙h top) ∙h right). + ``` + +2. Appending a homotopy `p : i ~ u` to the right gives a commuting square of + homotopies + + ```text + top + f ------------> g + | | + left | | right ∙h p + ∨ ∨ + h ------------> u. + bottom ∙h p + ``` + + More precisely, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ (left ∙h (bottom ∙h p) ~ top ∙h (right ∙h p)). + ``` + +3. Splicing a homotopy `p : h ~ u` and its inverse into the middle gives a + commuting square of homotopies + + ```text + top + f --------------> g + | | + left ∙h p | | right + ∨ ∨ + u --------------> i. + p⁻¹ ∙h bottom + ``` + + More precisely, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ ((left ∙h p) ∙h (p⁻¹ ∙h bottom) ~ top ∙h right). + ``` + + Similarly, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ ((left ∙h p⁻¹) ∙h (p ∙h bottom) ~ top ∙h right). + ``` + +4. Splicing a homotopy `p : g ~ u` and its inverse into the middle gives a + commuting square of homotopies + + ```text + top ∙h p + f --------> u + | | + left | | p⁻¹ ∙h right + ∨ ∨ + h --------> i. + bottom + ``` + + More precisely, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ (left ∙h bottom ~ (top ∙h p) ∙h (p⁻¹ ∙h right)). + ``` + + Similarly, we have an equivalence + + ```text + (left ∙h bottom ~ top ∙h right) ≃ (left ∙h bottom ~ (top ∙h p⁻¹) ∙h (p ∙h right)). + ``` + +These operations are useful in proofs involving homotopy algebra, because taking +`equiv-right-whisker-concat-coherence-square-homotopies` as an example, it +provides us with two maps: the forward direction states +`(p ∙h r ~ q ∙h s) → (p ∙h (r ∙h t)) ~ q ∙h (s ∙h t))`, which allows one to +append a homotopy without needing to reassociate on the right, and the backwards +direction conversely allows one to cancel out a homotopy in parentheses. + +#### Left whiskering coherences of commuting squares of homotopies + +For any homotopy `p : u ~ f` we obtain an equivalence ```text - top - f ------> g - | | - left | | right - v v - h ------> i - bottom + top p ∙h top + f -------> g u -------> g + | | | | + left | | right ≃ p ∙h left | | right + ∨ ∨ ∨ ∨ + h -------> i h -------> i + bottom bottom +``` + +of coherences of commuting squares of homotopies. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i u : (x : A) → B x} + where + + equiv-left-whisker-concat-coherence-square-homotopies : + (p : u ~ f) + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies (p ∙h top) (p ∙h left) right bottom + equiv-left-whisker-concat-coherence-square-homotopies + p top left right bottom = + equiv-Π-equiv-family + ( λ x → + equiv-left-whisker-concat-coherence-square-identifications + ( p x) (top x) (left x) (right x) (bottom x)) ``` -and consider a homotopy `H : e ~ f`. Then there is an equivalence of commuting -squares of homotopies +#### Right whiskering coherences of commuting squares of homotopies + +For any homotopy `p : i ~ u` we obtain an equivalence ```text - top H ∙h top - f ------> g e ----------> g - | | | | - left | | right ≃ H ∙h left | | right - ∨ ∨ ∨ ∨ - h ------> i h ----------> i - bottom bottom + top top + f -------> g f ------------> g + | | | | + left | | right ≃ left | | right ∙h p + ∨ ∨ ∨ ∨ + h -------> i h ------------> i + bottom bottom ∙h p ``` -This is the -{{#concept "double whiskering" Disambiguation="commuting squares of homotopies with respect to concatenation" Agda=double-whisker-coherence-square-homotopies}} -operation of commuting squares of homotopies with respect to concatenation. +of coherences of commuting squares of homotopies. ```agda module _ - { l1 l2 : Level} {A : UU l1} {B : UU l2} - { f g h h' k m : A → B} - ( H : f ~ g) {K : g ~ h} {K' : g ~ h'} {L : h ~ k} {L' : h' ~ k} (M : k ~ m) + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) where - equiv-double-whisker-coherence-square-homotopies : - ( coherence-square-homotopies K K' L L') ≃ - ( coherence-square-homotopies (H ∙h K) (H ∙h K') (L ∙h M) (L' ∙h M)) - equiv-double-whisker-coherence-square-homotopies = + equiv-right-whisker-concat-coherence-square-homotopies : + {u : (x : A) → B x} (p : i ~ u) → + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies top left (right ∙h p) (bottom ∙h p) + equiv-right-whisker-concat-coherence-square-homotopies p = equiv-Π-equiv-family - ( λ a → - equiv-double-whisker-square-identifications - ( H a) - ( K a) - ( K' a) - ( L a) - ( L' a) - ( M a)) - - double-whisker-coherence-square-homotopies : - ( coherence-square-homotopies K K' L L') → - ( coherence-square-homotopies (H ∙h K) (H ∙h K') (L ∙h M) (L' ∙h M)) - double-whisker-coherence-square-homotopies = - map-equiv equiv-double-whisker-coherence-square-homotopies - - double-unwhisker-concat-htpy-coherence-square-homotopies : - ( coherence-square-homotopies (H ∙h K) (H ∙h K') (L ∙h M) (L' ∙h M)) → - ( coherence-square-homotopies K K' L L') - double-unwhisker-concat-htpy-coherence-square-homotopies = - map-inv-equiv equiv-double-whisker-coherence-square-homotopies + ( λ x → + equiv-right-whisker-concat-coherence-square-identifications + ( top x) (left x) (right x) (bottom x) (p x)) ``` -### Whiskering a square of homotopies by a map +#### Left splicing coherences of commuting squares of homotopies -Given a square of homotopies +For any inverse pair of homotopies `p : g ~ u` and `q : u ~ g` equipped with +`α : inv-htpy p ~ q` we obtain an equivalence ```text - H - g -----> h - | | - H' | ⇗ | K - ∨ ∨ - h' -----> k - K' + top top + f -------> g f -----------> g + | | | | + left | | right ≃ left ∙h p | | right + ∨ ∨ ∨ ∨ + h -------> i u -----------> i + bottom q ∙h bottom +``` + +of coherences of commuting squares of homotopies. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + where + + equiv-left-splice-coherence-square-homotopies : + {u : (x : A) → B x} (p : h ~ u) (q : u ~ h) (α : inv-htpy p ~ q) → + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies top (left ∙h p) right (q ∙h bottom) + equiv-left-splice-coherence-square-homotopies p q α = + equiv-Π-equiv-family + ( λ x → + equiv-left-splice-coherence-square-identifications + ( top x) (left x) (right x) (bottom x) (p x) (q x) (α x)) ``` -and a map `f`, we may whisker it by a map on the left into a square of -homotopies +#### Right splicing coherences of commuting squares of homotopies + +For any inverse pair of homotopies `p : g ~ u` and `q : u ~ g` equipped with +`α : inv-htpy p ~ q` we obtain an equivalence ```text - f ·l H - fg --------> fh - | | - f ·l H' | ⇗ |f ·l K - ∨ ∨ - fh' --------> fk, - f ·l K' + top top ∙h p + f -------> g f --------> u + | | | | + left | | right ≃ left | | q ∙h right + ∨ ∨ ∨ ∨ + h -------> i h --------> i + bottom bottom ``` -and similarly we may whisker it on the right. +of coherences of commuting squares of homotopies. ```agda module _ - { l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} - ( f : B → C) {g h h' k : A → B} - ( H : g ~ h) (H' : g ~ h') {K : h ~ k} {K' : h' ~ k} + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) where - map-coherence-square-homotopies : - coherence-square-homotopies H H' K K' → - coherence-square-homotopies (f ·l H) (f ·l H') (f ·l K) (f ·l K') - map-coherence-square-homotopies α a = - map-coherence-square-identifications f (H a) (H' a) (K a) (K' a) (α a) + equiv-right-splice-coherence-square-homotopies : + {u : (x : A) → B x} (p : g ~ u) (q : u ~ g) (α : inv-htpy p ~ q) → + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies (top ∙h p) left (inv-htpy p ∙h right) bottom + equiv-right-splice-coherence-square-homotopies p q α = + equiv-Π-equiv-family + ( λ x → + equiv-right-splice-coherence-square-identifications + ( top x) (left x) (right x) (bottom x) (p x) (q x) (α x)) +``` + +### Double whiskering of commuting squares of homotopies +```agda module _ - { l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} - { g h h' k : B → C} (H : g ~ h) (H' : g ~ h') {K : h ~ k} {K' : h' ~ k} - ( f : A → B) + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h u v i : (x : A) → B x} where - right-whisker-comp-coherence-square-homotopies : - coherence-square-homotopies H H' K K' → - coherence-square-homotopies (H ·r f) (H' ·r f) (K ·r f) (K' ·r f) - right-whisker-comp-coherence-square-homotopies α = α ·r f + equiv-double-whisker-coherence-square-homotopies : + (p : f ~ g) + (top : g ~ u) (left : g ~ h) (right : u ~ v) (bottom : h ~ v) + (s : v ~ i) → + coherence-square-homotopies top left right bottom ≃ + coherence-square-homotopies + ( p ∙h top) + ( p ∙h left) + ( right ∙h s) + ( bottom ∙h s) + equiv-double-whisker-coherence-square-homotopies p top left right bottom q = + equiv-Π-equiv-family + ( λ x → + equiv-double-whisker-coherence-square-identifications + ( p x) (top x) (left x) (right x) (bottom x) (q x)) ``` diff --git a/src/foundation/commuting-squares-of-identifications.lagda.md b/src/foundation/commuting-squares-of-identifications.lagda.md index 8c537e89f2..75891f4367 100644 --- a/src/foundation/commuting-squares-of-identifications.lagda.md +++ b/src/foundation/commuting-squares-of-identifications.lagda.md @@ -2,21 +2,18 @@ ```agda module foundation.commuting-squares-of-identifications where + +open import foundation-core.commuting-squares-of-identifications public ```
Imports ```agda -open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.universe-levels -open import foundation.whiskering-identifications-concatenation open import foundation-core.equivalences -open import foundation-core.function-types open import foundation-core.identity-types -open import foundation-core.retractions -open import foundation-core.sections ```
@@ -42,197 +39,6 @@ identification is called a {{#concept "coherence" Disambiguation="commuting square of identifications" Agda=coherence-square-identifications}} of the square. -## Definitions - -### Commuting squares of identifications - -```agda -module _ - {l : Level} {A : UU l} {x y z w : A} - (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) - where - - coherence-square-identifications : UU l - coherence-square-identifications = left ∙ bottom ＝ top ∙ right -``` - -### Horizontally constant squares - -{{#concept "Horizontally constant squares" Disambiguation="identifications" Agda=horizontal-refl-coherence-square-identifications}} -are commuting squares of identifications of the form - -```text - refl - a -----> a - | | - p | | p - ∨ ∨ - b -----> b. - refl -``` - -```agda -module _ - {l : Level} {A : UU l} {a b : A} (p : a ＝ b) - where - - horizontal-refl-coherence-square-identifications : - coherence-square-identifications refl p p refl - horizontal-refl-coherence-square-identifications = right-unit -``` - -### Vertically constant squares - -{{#concept "Vertically constant squares" Disambiguation="identifications" Agda=vertical-refl-coherence-square-identifications}} -are commuting squares of identifications of the form - -```text - p - a -----> b - | | - refl | | refl - ∨ ∨ - a -----> b. - p -``` - -```agda -module _ - {l : Level} {A : UU l} {a b : A} (p : a ＝ b) - where - - vertical-refl-coherence-square-identifications : - coherence-square-identifications p refl refl p - vertical-refl-coherence-square-identifications = inv right-unit -``` - -## Operations - -### Inverting squares of identifications horizontally - -Given a commuting square of identifications - -```text - top - x -------> y - | | - left | | right - ∨ ∨ - z -------> w, - bottom -``` - -the square of identifications - -```text - inv top - y ------------> x - | | - right | | left - ∨ ∨ - w ------------> z - inv bottom -``` - -commutes. - -```agda -module _ - {l : Level} {A : UU l} {x y z w : A} - where - - horizontal-inv-coherence-square-identifications : - (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → - coherence-square-identifications top left right bottom → - coherence-square-identifications (inv top) right left (inv bottom) - horizontal-inv-coherence-square-identifications refl refl right refl coh = - right-unit ∙ inv coh -``` - -### Inverting squares of identifications vertically - -Given a commuting square of identifications - -```text - top - x -------> y - | | - left | | right - ∨ ∨ - z -------> w, - bottom -``` - -the square of identifications - -```text - bottom - z -------> w - | | - inv left | | inv right - ∨ ∨ - x -------> y - top -``` - -commutes. - -```agda -module _ - {l : Level} {A : UU l} {x y z w : A} - where - - vertical-inv-coherence-square-identifications : - (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → - coherence-square-identifications top left right bottom → - coherence-square-identifications bottom (inv left) (inv right) top - vertical-inv-coherence-square-identifications refl refl refl refl refl = refl -``` - -### Functions acting on squares of identifications - -Given a commuting square of identifications - -```text - top - x -------> y - | | - left | | right - ∨ ∨ - z -------> w - bottom -``` - -in a type `A`, and given a map `f : A → B`, the square of identifications - -```text - ap f top - f x -----------> f y - | | - ap f left | | ap f right - ∨ ∨ - z -------------> w - ap f bottom -``` - -commutes. - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} {x y z w : A} (f : A → B) - where - - map-coherence-square-identifications : - (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → - coherence-square-identifications top left right bottom → - coherence-square-identifications - ( ap f top) - ( ap f left) - ( ap f right) - ( ap f bottom) - map-coherence-square-identifications refl refl right refl coh = ap (ap f) coh -``` - ### Concatenating identifications of edges and coherences of commuting squares of identifications Consider a commuting square of identifications and an identification of one of @@ -252,10 +58,6 @@ Then any identification witnessing that the square commutes can be concatenated with the identification on the side, to obtain a new commuting square of identifications. -**Note.** To avoid cyclic module dependencies we will give direct proofs that -concatenating identifications of edges of a square with the coherence of its -commutativity is an equivalence. - #### Concatenating identifications of the top edge with a coherence of a commuting square of identifications Consider a commuting diagram of identifications @@ -290,46 +92,26 @@ module _ {top' : x ＝ y} (s : top ＝ top') where - concat-top-identification-coherence-square-identifications : - coherence-square-identifications top left right bottom → - coherence-square-identifications top' left right bottom - concat-top-identification-coherence-square-identifications t = - t ∙ ap (concat' _ right) s - - inv-concat-top-identification-coherence-square-identifications : - coherence-square-identifications top' left right bottom → - coherence-square-identifications top left right bottom - inv-concat-top-identification-coherence-square-identifications t = - t ∙ inv (ap (concat' _ right) s) - - is-section-inv-concat-top-identification-coherence-square-identifications : - is-section - concat-top-identification-coherence-square-identifications - inv-concat-top-identification-coherence-square-identifications - is-section-inv-concat-top-identification-coherence-square-identifications = - is-section-inv-concat' (ap (concat' _ right) s) - - is-retraction-inv-concat-top-identification-coherence-square-identifications : - is-retraction - concat-top-identification-coherence-square-identifications - inv-concat-top-identification-coherence-square-identifications - is-retraction-inv-concat-top-identification-coherence-square-identifications = - is-retraction-inv-concat' (ap (concat' _ right) s) - abstract is-equiv-concat-top-identification-coherence-square-identifications : - is-equiv concat-top-identification-coherence-square-identifications + is-equiv + ( concat-top-identification-coherence-square-identifications + top left right bottom s) is-equiv-concat-top-identification-coherence-square-identifications = is-equiv-is-invertible - inv-concat-top-identification-coherence-square-identifications - is-section-inv-concat-top-identification-coherence-square-identifications - is-retraction-inv-concat-top-identification-coherence-square-identifications + ( inv-concat-top-identification-coherence-square-identifications + top left right bottom s) + ( is-section-inv-concat-top-identification-coherence-square-identifications + top left right bottom s) + ( is-retraction-inv-concat-top-identification-coherence-square-identifications + top left right bottom s) equiv-concat-top-identification-coherence-square-identifications : coherence-square-identifications top left right bottom ≃ coherence-square-identifications top' left right bottom pr1 equiv-concat-top-identification-coherence-square-identifications = concat-top-identification-coherence-square-identifications + top left right bottom s pr2 equiv-concat-top-identification-coherence-square-identifications = is-equiv-concat-top-identification-coherence-square-identifications ``` @@ -367,45 +149,26 @@ module _ {left' : x ＝ z} (s : left ＝ left') where - concat-left-identification-coherence-square-identifications : - coherence-square-identifications top left right bottom → - coherence-square-identifications top left' right bottom - concat-left-identification-coherence-square-identifications t = - inv (ap (concat' _ bottom) s) ∙ t - - inv-concat-left-identification-coherence-square-identifications : - coherence-square-identifications top left' right bottom → - coherence-square-identifications top left right bottom - inv-concat-left-identification-coherence-square-identifications t = - ap (concat' _ bottom) s ∙ t - - is-section-inv-concat-left-identification-coherence-square-identifications : - is-section - concat-left-identification-coherence-square-identifications - inv-concat-left-identification-coherence-square-identifications - is-section-inv-concat-left-identification-coherence-square-identifications = - is-retraction-inv-concat (ap (concat' _ bottom) s) - - is-retraction-inv-concat-left-identification-coherence-square-identifications : - is-retraction - concat-left-identification-coherence-square-identifications - inv-concat-left-identification-coherence-square-identifications - is-retraction-inv-concat-left-identification-coherence-square-identifications = - is-section-inv-concat (ap (concat' _ bottom) s) - - is-equiv-concat-left-identification-coherence-square-identifications : - is-equiv concat-left-identification-coherence-square-identifications - is-equiv-concat-left-identification-coherence-square-identifications = - is-equiv-is-invertible - inv-concat-left-identification-coherence-square-identifications - is-section-inv-concat-left-identification-coherence-square-identifications - is-retraction-inv-concat-left-identification-coherence-square-identifications + abstract + is-equiv-concat-left-identification-coherence-square-identifications : + is-equiv + ( concat-left-identification-coherence-square-identifications + top left right bottom s) + is-equiv-concat-left-identification-coherence-square-identifications = + is-equiv-is-invertible + ( inv-concat-left-identification-coherence-square-identifications + top left right bottom s) + ( is-section-inv-concat-left-identification-coherence-square-identifications + top left right bottom s) + ( is-retraction-inv-concat-left-identification-coherence-square-identifications + top left right bottom s) equiv-concat-left-identification-coherence-square-identifications : coherence-square-identifications top left right bottom ≃ coherence-square-identifications top left' right bottom pr1 equiv-concat-left-identification-coherence-square-identifications = concat-left-identification-coherence-square-identifications + top left right bottom s pr2 equiv-concat-left-identification-coherence-square-identifications = is-equiv-concat-left-identification-coherence-square-identifications ``` @@ -443,46 +206,26 @@ module _ {right' : y ＝ w} (s : right ＝ right') where - concat-right-identification-coherence-square-identifications : - coherence-square-identifications top left right bottom → - coherence-square-identifications top left right' bottom - concat-right-identification-coherence-square-identifications t = - t ∙ ap (concat top _) s - - inv-concat-right-identification-coherence-square-identifications : - coherence-square-identifications top left right' bottom → - coherence-square-identifications top left right bottom - inv-concat-right-identification-coherence-square-identifications t = - t ∙ inv (ap (concat top _) s) - - is-section-inv-concat-right-identification-coherence-square-identifications : - is-section - concat-right-identification-coherence-square-identifications - inv-concat-right-identification-coherence-square-identifications - is-section-inv-concat-right-identification-coherence-square-identifications = - is-section-inv-concat' (ap (concat top _) s) - - is-retraction-inv-concat-right-identification-coherence-square-identifications : - is-retraction - concat-right-identification-coherence-square-identifications - inv-concat-right-identification-coherence-square-identifications - is-retraction-inv-concat-right-identification-coherence-square-identifications = - is-retraction-inv-concat' (ap (concat top _) s) - abstract is-equiv-concat-right-identification-coherence-square-identifications : - is-equiv concat-right-identification-coherence-square-identifications + is-equiv + ( concat-right-identification-coherence-square-identifications + top left right bottom s) is-equiv-concat-right-identification-coherence-square-identifications = is-equiv-is-invertible - inv-concat-right-identification-coherence-square-identifications - is-section-inv-concat-right-identification-coherence-square-identifications - is-retraction-inv-concat-right-identification-coherence-square-identifications + ( inv-concat-right-identification-coherence-square-identifications + top left right bottom s) + ( is-section-inv-concat-right-identification-coherence-square-identifications + top left right bottom s) + ( is-retraction-inv-concat-right-identification-coherence-square-identifications + top left right bottom s) equiv-concat-right-identification-coherence-square-identifications : coherence-square-identifications top left right bottom ≃ coherence-square-identifications top left right' bottom pr1 equiv-concat-right-identification-coherence-square-identifications = concat-right-identification-coherence-square-identifications + top left right bottom s pr2 equiv-concat-right-identification-coherence-square-identifications = is-equiv-concat-right-identification-coherence-square-identifications ``` @@ -521,45 +264,25 @@ module _ {bottom' : z ＝ w} (s : bottom ＝ bottom') where - concat-bottom-identification-coherence-square-identifications : - coherence-square-identifications top left right bottom → - coherence-square-identifications top left right bottom' - concat-bottom-identification-coherence-square-identifications t = - inv (ap (concat left _) s) ∙ t - - inv-concat-bottom-identification-coherence-square-identifications : - coherence-square-identifications top left right bottom' → - coherence-square-identifications top left right bottom - inv-concat-bottom-identification-coherence-square-identifications t = - ap (concat left _) s ∙ t - - is-section-inv-concat-bottom-identification-coherence-square-identifications : - is-section - concat-bottom-identification-coherence-square-identifications - inv-concat-bottom-identification-coherence-square-identifications - is-section-inv-concat-bottom-identification-coherence-square-identifications = - is-retraction-inv-concat (ap (concat left _) s) - - is-retraction-inv-concat-bottom-identification-coherence-square-identifications : - is-retraction - concat-bottom-identification-coherence-square-identifications - inv-concat-bottom-identification-coherence-square-identifications - is-retraction-inv-concat-bottom-identification-coherence-square-identifications = - is-section-inv-concat (ap (concat left _) s) - is-equiv-concat-bottom-identification-coherence-square-identifications : - is-equiv concat-bottom-identification-coherence-square-identifications + is-equiv + ( concat-bottom-identification-coherence-square-identifications + top left right bottom s) is-equiv-concat-bottom-identification-coherence-square-identifications = is-equiv-is-invertible - inv-concat-bottom-identification-coherence-square-identifications - is-section-inv-concat-bottom-identification-coherence-square-identifications - is-retraction-inv-concat-bottom-identification-coherence-square-identifications + ( inv-concat-bottom-identification-coherence-square-identifications + top left right bottom s) + ( is-section-inv-concat-bottom-identification-coherence-square-identifications + top left right bottom s) + ( is-retraction-inv-concat-bottom-identification-coherence-square-identifications + top left right bottom s) equiv-concat-bottom-identification-coherence-square-identifications : coherence-square-identifications top left right bottom ≃ coherence-square-identifications top left right bottom' pr1 equiv-concat-bottom-identification-coherence-square-identifications = concat-bottom-identification-coherence-square-identifications + top left right bottom s pr2 equiv-concat-bottom-identification-coherence-square-identifications = is-equiv-concat-bottom-identification-coherence-square-identifications ``` @@ -668,7 +391,7 @@ we may consider four ways of attaching new identifications to it: ``` These operations are useful in proofs involving path algebra, because taking -`equiv-right-whisker-concat-coherence-square-identicications` as an example, it +`equiv-right-whisker-concat-coherence-square-identifications` as an example, it provides us with two maps: the forward direction states `(p ∙ r ＝ q ∙ s) → (p ∙ (r ∙ t)) ＝ q ∙ (s ∙ t))`, which allows one to append an identification without needing to reassociate on the right, and the backwards @@ -700,27 +423,9 @@ module _ (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → coherence-square-identifications top left right bottom ≃ coherence-square-identifications (p ∙ top) (p ∙ left) right bottom - equiv-left-whisker-concat-coherence-square-identifications refl - top left right bottom = - id-equiv - - left-whisker-concat-coherence-square-identifications : - (p : u ＝ x) - (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → - coherence-square-identifications top left right bottom → - coherence-square-identifications (p ∙ top) (p ∙ left) right bottom - left-whisker-concat-coherence-square-identifications + equiv-left-whisker-concat-coherence-square-identifications refl top left right bottom = - id - - left-unwhisker-concat-coherence-square-identifications : - (p : u ＝ x) - (top : x ＝ y) (left : x ＝ z) (right : y ＝ w) (bottom : z ＝ w) → - coherence-square-identifications (p ∙ top) (p ∙ left) right bottom → - coherence-square-identifications top left right bottom - left-unwhisker-concat-coherence-square-identifications - refl top left right bottom = - id + id-equiv ``` #### Right whiskering coherences of commuting squares of identifications @@ -762,43 +467,6 @@ module _ ( right) ( bottom) ( inv right-unit)) - - right-whisker-concat-coherence-square-identifications : - coherence-square-identifications top left right bottom → - {u : A} (p : w ＝ u) → - coherence-square-identifications top left (right ∙ p) (bottom ∙ p) - right-whisker-concat-coherence-square-identifications s refl = - concat-bottom-identification-coherence-square-identifications - ( top) - ( left) - ( right ∙ refl) - ( bottom) - ( inv right-unit) - ( concat-right-identification-coherence-square-identifications - ( top) - ( left) - ( right) - ( bottom) - ( inv right-unit) - ( s)) - - right-unwhisker-cohernece-square-identifications : - {u : A} (p : w ＝ u) → - coherence-square-identifications top left (right ∙ p) (bottom ∙ p) → - coherence-square-identifications top left right bottom - right-unwhisker-cohernece-square-identifications refl = - ( inv-concat-right-identification-coherence-square-identifications - ( top) - ( left) - ( right) - ( bottom) - ( inv right-unit)) ∘ - ( inv-concat-bottom-identification-coherence-square-identifications - ( top) - ( left) - ( right ∙ refl) - ( bottom) - ( inv right-unit)) ``` #### Left splicing coherences of commuting squares of identifications @@ -835,30 +503,6 @@ module _ ( right) ( bottom) ( inv right-unit) - - left-splice-coherence-square-identifications : - {u : A} (p : z ＝ u) (q : u ＝ z) (α : inv p ＝ q) → - coherence-square-identifications top left right bottom → - coherence-square-identifications top (left ∙ p) right (q ∙ bottom) - left-splice-coherence-square-identifications refl .refl refl = - concat-left-identification-coherence-square-identifications - ( top) - ( left) - ( right) - ( bottom) - ( inv right-unit) - - left-unsplice-coherence-square-identifications : - {u : A} (p : z ＝ u) (q : u ＝ z) (α : inv p ＝ q) → - coherence-square-identifications top (left ∙ p) right (q ∙ bottom) → - coherence-square-identifications top left right bottom - left-unsplice-coherence-square-identifications refl .refl refl = - inv-concat-left-identification-coherence-square-identifications - ( top) - ( left) - ( right) - ( bottom) - ( inv right-unit) ``` #### Right splicing coherences of commuting squares of identifications @@ -895,30 +539,6 @@ module _ ( right) ( bottom) ( inv right-unit) - - right-splice-coherence-square-identifications : - {u : A} (p : y ＝ u) (q : u ＝ y) (α : inv p ＝ q) → - coherence-square-identifications top left right bottom → - coherence-square-identifications (top ∙ p) left (inv p ∙ right) bottom - right-splice-coherence-square-identifications refl .refl refl = - concat-top-identification-coherence-square-identifications - ( top) - ( left) - ( right) - ( bottom) - ( inv right-unit) - - right-unsplice-coherence-square-identifications : - {u : A} (p : y ＝ u) (q : u ＝ y) (α : inv p ＝ q) → - coherence-square-identifications (top ∙ p) left (inv p ∙ right) bottom → - coherence-square-identifications top left right bottom - right-unsplice-coherence-square-identifications refl .refl refl = - inv-concat-top-identification-coherence-square-identifications - ( top) - ( left) - ( right) - ( bottom) - ( inv right-unit) ``` ### Double whiskering of commuting squares of identifications @@ -928,7 +548,7 @@ module _ {l : Level} {A : UU l} {x y z u v w : A} where - equiv-double-whisker-square-identifications : + equiv-double-whisker-coherence-square-identifications : (p : x ＝ y) (top : y ＝ u) (left : y ＝ z) (right : u ＝ v) (bottom : z ＝ v) (s : v ＝ w) → @@ -938,7 +558,8 @@ module _ ( p ∙ left) ( right ∙ s) ( bottom ∙ s) - equiv-double-whisker-square-identifications p top left right bottom q = + equiv-double-whisker-coherence-square-identifications + p top left right bottom q = equiv-left-whisker-concat-coherence-square-identifications p top left ( right ∙ q) ( bottom ∙ q) ∘e @@ -949,734 +570,3 @@ module _ ( bottom) ( q) ``` - -### Horizontally pasting squares of identifications - -Consider two squares of identifications as in the diagram - -```text - top-left top-right - a -------------> b -------------> c - | | | - left | | middle | right - ∨ ∨ ∨ - d -------------> e -------------> f - bottom-left bottom-right -``` - -with `s : left ∙ bottom-left ＝ top-left ∙ middle` and t : middle ∙ bottom-right -＝ top-right ∙ right`. Then the outer square commutes. - -```agda -module _ - {l : Level} {A : UU l} {a b c d e f : A} - (top-left : a ＝ b) (top-right : b ＝ c) - (left : a ＝ d) (middle : b ＝ e) (right : c ＝ f) - (bottom-left : d ＝ e) (bottom-right : e ＝ f) - where - - horizontal-pasting-coherence-square-identifications : - coherence-square-identifications top-left left middle bottom-left → - coherence-square-identifications top-right middle right bottom-right → - coherence-square-identifications - (top-left ∙ top-right) left right (bottom-left ∙ bottom-right) - horizontal-pasting-coherence-square-identifications s t = - ( right-whisker-concat-coherence-square-identifications - ( top-left) - ( left) - ( middle) - ( bottom-left) - ( s) - ( bottom-right)) ∙ - ( ( inv (assoc top-left middle bottom-right)) ∙ - ( left-whisker-concat-coherence-square-identifications - ( top-left) - ( top-right) - ( middle) - ( right) - ( bottom-right) - ( t))) -``` - -### Vertically pasting squares of identifications - -Consider two squares of identifications as in the diagram - -```text - top - a --------> b - | | - top-left | | top-right - ∨ middle ∨ - c --------> d - | | - bottom-left | | bottom-right - ∨ ∨ - e --------> f - bottom -``` - -with `s : top-left ∙ middle ＝ top ∙ top-right` and -`t : bottom-left ∙ bottom ＝ middle ∙ bottom-right`. Then the outer square -commutes. - -```agda -module _ - {l : Level} {A : UU l} {a b c d e f : A} - (top : a ＝ b) (top-left : a ＝ c) (top-right : b ＝ d) - (middle : c ＝ d) (bottom-left : c ＝ e) (bottom-right : d ＝ f) - (bottom : e ＝ f) - where - - vertical-pasting-coherence-square-identifications : - coherence-square-identifications top top-left top-right middle → - coherence-square-identifications middle bottom-left bottom-right bottom → - coherence-square-identifications - top (top-left ∙ bottom-left) (top-right ∙ bottom-right) bottom - vertical-pasting-coherence-square-identifications p q = - ( left-whisker-concat-coherence-square-identifications - ( top-left) - ( middle) - ( bottom-left) - ( bottom-right) - ( bottom) - ( q)) ∙ - ( ( assoc top-left middle bottom-right) ∙ - ( right-whisker-concat-coherence-square-identifications - ( top) - ( top-left) - ( top-right) - ( middle) - ( p) - ( bottom-right))) -``` - -## Properties - -### Left unit law for horizontal pasting of commuting squares of identifications - -```agda -module _ - {l : Level} {A : UU l} {a b c d : A} - where - - left-unit-law-horizontal-pasting-coherence-square-identifications : - (top : a ＝ b) (left : a ＝ c) (right : b ＝ d) (bottom : c ＝ d) - (s : coherence-square-identifications top left right bottom) → - horizontal-pasting-coherence-square-identifications - ( refl) - ( top) - ( left) - ( left) - ( right) - ( refl) - ( bottom) - ( horizontal-refl-coherence-square-identifications left) - ( s) ＝ - s - left-unit-law-horizontal-pasting-coherence-square-identifications - refl refl right refl s = refl -``` - -### Right unit law for horizontal pasting of commuting squares of identifications - -```agda -module _ - {l : Level} {A : UU l} {a b c d : A} - where - - right-unit-law-horizontal-pasting-coherence-square-identifications : - (top : a ＝ b) (left : a ＝ c) (right : b ＝ d) (bottom : c ＝ d) - (s : coherence-square-identifications top left right bottom) → - horizontal-pasting-coherence-square-identifications - ( top) - ( refl) - ( left) - ( right) - ( right) - ( bottom) - ( refl) - ( s) - ( horizontal-refl-coherence-square-identifications right) ∙ - right-whisker-concat right-unit right ＝ - left-whisker-concat left right-unit ∙ s - right-unit-law-horizontal-pasting-coherence-square-identifications - refl refl .refl refl refl = - refl -``` - -### Left unit law for vertical pasting of commuting squares of identifications - -```agda -module _ - {l : Level} {A : UU l} {a b c d : A} - where - - left-unit-law-vertical-pasting-coherence-square-identifications : - (top : a ＝ b) (left : a ＝ c) (right : b ＝ d) (bottom : c ＝ d) - (s : coherence-square-identifications top left right bottom) → - vertical-pasting-coherence-square-identifications - ( top) - ( refl) - ( refl) - ( top) - ( left) - ( right) - ( bottom) - ( vertical-refl-coherence-square-identifications top) - ( s) ＝ - s - left-unit-law-vertical-pasting-coherence-square-identifications - refl refl .refl refl refl = refl -``` - -### Right unit law for vertical pasting of commuting squares of identifications - -```agda -module _ - {l : Level} {A : UU l} {a b c d : A} - where - - right-unit-law-vertical-pasting-coherence-square-identifications : - (top : a ＝ b) (left : a ＝ c) (right : b ＝ d) (bottom : c ＝ d) - (s : coherence-square-identifications top left right bottom) → - vertical-pasting-coherence-square-identifications - ( top) - ( left) - ( right) - ( bottom) - ( refl) - ( refl) - ( bottom) - ( s) - ( vertical-refl-coherence-square-identifications bottom) ∙ - left-whisker-concat top right-unit ＝ - right-whisker-concat right-unit bottom ∙ s - right-unit-law-vertical-pasting-coherence-square-identifications - refl refl .(refl ∙ refl) refl refl = - refl -``` - -### Computing the right whiskering of a vertically constant square with an identification - -Consider the vertically constant square of identifications - -```text - p - x -----> y - | | - refl | | refl - ∨ ∨ - x -----> y - p -``` - -at an identification `p : x ＝ y`, and consider an identification `q : y ＝ z`. -Then the right whiskering of the above square with `q` is the commuting square -of identifications - -```text - p - x -------> y - | | - refl | refl | q - ∨ ∨ - x -------> z - p ∙ q -``` - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - right-whisker-concat-vertical-refl-coherence-square-identifications : - {x y z : A} (p : x ＝ y) (q : y ＝ z) → - right-whisker-concat-coherence-square-identifications p refl refl p - ( vertical-refl-coherence-square-identifications p) - ( q) ＝ - refl - right-whisker-concat-vertical-refl-coherence-square-identifications - refl refl = - refl -``` - -### Computing the right whiskering of a horizontally constant square with an identification - -Consider a horizontally constant commuting square of identifications - -```text - refl - x -----> x - | | - p | | p - ∨ ∨ - y -----> y - refl -``` - -at an identification `p` and consider an identification `q : y ＝ z`. Then the -right whiskering of the above square with `q` is the square - -```text - refl - x -----> x - | | - p | refl | p ∙ q - ∨ ∨ - y -----> z. - q -``` - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - right-whisker-concat-horizontal-refl-coherence-square-identifications : - {x y z : A} (p : x ＝ y) (q : y ＝ z) → - right-whisker-concat-coherence-square-identifications refl p p refl - ( horizontal-refl-coherence-square-identifications p) - ( q) ＝ - refl - right-whisker-concat-horizontal-refl-coherence-square-identifications - refl refl = - refl -``` - -### Computing the left whiskering of a horizontally constant square with an identification - -Consider an identification `p : x ＝ y` and a horizontally constant commuting -square of identifications - -```text - refl - y -----> y - | | - q | | q - ∨ ∨ - z -----> z - refl -``` - -at an identification `q : y ＝ z`. The the left whiskering of the above square -with `p` is the commuting square - -```text - q ∙ refl - x ------------------------------------------------------> y - | | - q ∙ p | right-unit ∙ inv (right-whisker-concat right-unit p) | p - ∨ ∨ - z ------------------------------------------------------> z. - refl -``` - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - left-whisker-concat-horizontal-refl-coherence-square-identifications : - {x y z : A} (p : x ＝ y) (q : y ＝ z) → - left-whisker-concat-coherence-square-identifications p refl q q refl - ( horizontal-refl-coherence-square-identifications q) ∙ - right-whisker-concat right-unit q ＝ - right-unit - left-whisker-concat-horizontal-refl-coherence-square-identifications - refl refl = - refl - - left-whisker-concat-horizontal-refl-coherence-square-identifications' : - {x y z : A} (p : x ＝ y) (q : y ＝ z) → - left-whisker-concat-coherence-square-identifications p refl q q refl - ( horizontal-refl-coherence-square-identifications q) ＝ - right-unit ∙ inv (right-whisker-concat right-unit q) - left-whisker-concat-horizontal-refl-coherence-square-identifications' - refl refl = - refl -``` - -### Computing the left whiskering of a vertically constant square with an identification - -Consider the vertically constant square of identifications - -```text - q - y -----> z - | | - refl | | refl - ∨ ∨ - y -----> z - q -``` - -at an identification `q : y ＝ z` and consider an identification `p : x ＝ y`. -Then the left whiskering of the above square with `p` is the square - -```text - p ∙ q - x ---------------------------------------------------> z - | | - p ∙ refl | right-whisker-concat right-unit q ∙ inv right-unit | refl - ∨ ∨ - y ---------------------------------------------------> z. - q -``` - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - left-whisker-concat-vertical-refl-coherence-square-identifications : - {x y z : A} (p : x ＝ y) (q : y ＝ z) → - left-whisker-concat-coherence-square-identifications p q refl refl q - ( vertical-refl-coherence-square-identifications q) ∙ - right-unit ＝ - right-whisker-concat right-unit q - left-whisker-concat-vertical-refl-coherence-square-identifications - refl refl = - refl - - left-whisker-concat-vertical-refl-coherence-square-identifications' : - {x y z : A} (p : x ＝ y) (q : y ＝ z) → - left-whisker-concat-coherence-square-identifications p q refl refl q - ( vertical-refl-coherence-square-identifications q) ＝ - right-whisker-concat right-unit q ∙ inv right-unit - left-whisker-concat-vertical-refl-coherence-square-identifications' - refl refl = - refl -``` - -### Left whiskering horizontal concatenations of squares with identifications - -Consider a commuting diagram of identifications of the form - -```text - top-left top-right - a -------------> c -------------> e - | | | - left | | middle | right - ∨ ∨ ∨ - b -------------> d -------------> f - bottom-left bottom-right -``` - -and consider an identification `p : x ＝ a`. Then the left whiskering of `p` and -the horizontal concatenation of coherences of commuting squares is up to -associativity the horizontal concatenation of the squares - -```text - p ∙ top-left top-right - x -------------> c -------------> e - | | | - p ∙ left | | middle | right - ∨ ∨ ∨ - b -------------> d -------------> f - bottom-left bottom-right -``` - -where the left square is the left whiskering of `p` and the original left -square. - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - left-whisker-concat-horizontal-pasting-coherence-square-identifications : - {x a b c d e f : A} (p : x ＝ a) - (top-left : a ＝ c) (top-right : c ＝ e) - (left : a ＝ b) (middle : c ＝ d) (right : e ＝ f) - (bottom-left : b ＝ d) (bottom-right : d ＝ f) - (l : coherence-square-identifications top-left left middle bottom-left) - (r : coherence-square-identifications top-right middle right bottom-right) → - left-whisker-concat-coherence-square-identifications p - ( top-left ∙ top-right) - ( left) - ( right) - ( bottom-left ∙ bottom-right) - ( horizontal-pasting-coherence-square-identifications - ( top-left) - ( top-right) - ( left) - ( middle) - ( right) - ( bottom-left) - ( bottom-right) - ( l) - ( r)) ＝ - horizontal-pasting-coherence-square-identifications - ( p ∙ top-left) - ( top-right) - ( p ∙ left) - ( middle) - ( right) - ( bottom-left) - ( bottom-right) - ( left-whisker-concat-coherence-square-identifications p - ( top-left) - ( left) - ( middle) - ( bottom-left) - ( l)) - ( r) ∙ - right-whisker-concat - ( assoc p top-left top-right) - ( right) - left-whisker-concat-horizontal-pasting-coherence-square-identifications - refl top-left top-right left middle right bottom-left bottom-right l r = - inv right-unit -``` - -### Left whiskering vertical concatenations of squares with identifications - -Consider two squares of identifications as in the diagram - -```text - top - a --------> b - | | - top-left | | top-right - ∨ middle ∨ - c --------> d - | | - bottom-left | | bottom-right - ∨ ∨ - e --------> f - bottom -``` - -and consider an identification `p : x ＝ a`. Then the left whiskering of `p` -with the vertical pasting of the two squares above is up to associativity the -vertical pasting of the squares - -```text - p ∙ top - x --------> b - | | - p ∙ top-left | | top-right - ∨ middle ∨ - c --------> d - | | - bottom-left | | bottom-right - ∨ ∨ - e --------> f. - bottom -``` - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - left-whisker-concat-vertical-concat-coherence-square-identifications : - {x a b c d e f : A} (p : x ＝ a) → - (top : a ＝ b) (top-left : a ＝ c) (top-right : b ＝ d) (middle : c ＝ d) - (bottom-left : c ＝ e) (bottom-right : d ＝ f) (bottom : e ＝ f) - (t : coherence-square-identifications top top-left top-right middle) → - (b : - coherence-square-identifications middle bottom-left bottom-right bottom) → - right-whisker-concat (assoc p top-left bottom-left) bottom ∙ - left-whisker-concat-coherence-square-identifications p - ( top) - ( top-left ∙ bottom-left) - ( top-right ∙ bottom-right) - ( bottom) - ( vertical-pasting-coherence-square-identifications - ( top) - ( top-left) - ( top-right) - ( middle) - ( bottom-left) - ( bottom-right) - ( bottom) - ( t) - ( b)) ＝ - vertical-pasting-coherence-square-identifications - ( p ∙ top) - ( p ∙ top-left) - ( top-right) - ( middle) - ( bottom-left) - ( bottom-right) - ( bottom) - ( left-whisker-concat-coherence-square-identifications p - ( top) - ( top-left) - ( top-right) - ( middle) - ( t)) - ( b) - left-whisker-concat-vertical-concat-coherence-square-identifications - refl top top-left top-right middle bottom-left bottom-right bottom t b = - refl -``` - -### Right whiskering horizontal pastings of commuting squares of identifications - -Consider a commuting diagram of identifications of the form - -```text - top-left top-right - a -------------> c -------------> e - | | | - left | | middle | right - ∨ ∨ ∨ - b -------------> d -------------> f - bottom-left bottom-right -``` - -and consider an identification `q : f ＝ y`. Then the right whiskering of the -horizontal pasting of the squares above is up to associativity the horizontal -pasting of the squares - -```text - top-left top-right - a -------------> c ------------------> e - | | | - left | | middle | right ∙ q - ∨ ∨ ∨ - b -------------> d ------------------> y - bottom-left bottom-right ∙ q -``` - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - right-whisker-concat-horizontal-pasting-coherence-square-identifications : - {a b c d e f y : A} - (top-left : a ＝ c) (top-right : c ＝ e) - (left : a ＝ b) (middle : c ＝ d) (right : e ＝ f) - (bottom-left : b ＝ d) (bottom-right : d ＝ f) - (l : coherence-square-identifications top-left left middle bottom-left) → - (r : coherence-square-identifications top-right middle right bottom-right) → - (q : f ＝ y) → - right-whisker-concat-coherence-square-identifications - ( top-left ∙ top-right) - ( left) - ( right) - ( bottom-left ∙ bottom-right) - ( horizontal-pasting-coherence-square-identifications - ( top-left) - ( top-right) - ( left) - ( middle) - ( right) - ( bottom-left) - ( bottom-right) - ( l) - ( r)) - ( q) ＝ - left-whisker-concat left (assoc bottom-left bottom-right q) ∙ - horizontal-pasting-coherence-square-identifications - ( top-left) - ( top-right) - ( left) - ( middle) - ( right ∙ q) - ( bottom-left) - ( bottom-right ∙ q) - ( l) - ( right-whisker-concat-coherence-square-identifications - ( top-right) - ( middle) - ( right) - ( bottom-right) - ( r) - ( q)) - right-whisker-concat-horizontal-pasting-coherence-square-identifications - refl refl refl .refl .refl refl refl refl refl refl = - refl -``` - -### Right whiskering vertical concatenations of squares with identifications - -Consider two squares of identifications as in the diagram - -```text - top - a --------> b - | | - top-left | | top-right - ∨ middle ∨ - c --------> d - | | - bottom-left | | bottom-right - ∨ ∨ - e --------> f - bottom -``` - -and consider an identification `q : f ＝ y`. Then the right whiskering of the -vertical pasting of the two squares above with `q` is up to associativity the -vertical pasting of the squares - -```text - top - a ------------> b - | | - top-left | | top-right - ∨ middle ∨ - c ------------> d - | | - bottom-left | | bottom-right ∙ q - ∨ ∨ - e ------------> y. - bottom ∙ q -``` - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - right-whisker-concat-vertical-pasting-coherence-square-identifications : - {a b c d e f y : A} - (top : a ＝ b) (top-left : a ＝ c) (top-right : b ＝ d) - (middle : c ＝ d) - (bottom-left : c ＝ e) (bottom-right : d ＝ f) (bottom : e ＝ f) - (t : coherence-square-identifications top top-left top-right middle) → - (b : - coherence-square-identifications middle bottom-left bottom-right bottom) → - (q : f ＝ y) → - right-whisker-concat-coherence-square-identifications - ( top) - ( top-left ∙ bottom-left) - ( top-right ∙ bottom-right) - ( bottom) - ( vertical-pasting-coherence-square-identifications - ( top) - ( top-left) - ( top-right) - ( middle) - ( bottom-left) - ( bottom-right) - ( bottom) - ( t) - ( b)) - ( q) ∙ - left-whisker-concat top (assoc top-right bottom-right q) ＝ - vertical-pasting-coherence-square-identifications - ( top) - ( top-left) - ( top-right) - ( middle) - ( bottom-left) - ( bottom-right ∙ q) - ( bottom ∙ q) - ( t) - ( right-whisker-concat-coherence-square-identifications - ( middle) - ( bottom-left) - ( bottom-right) - ( bottom) - ( b) - ( q)) - right-whisker-concat-vertical-pasting-coherence-square-identifications - refl refl .refl refl refl .refl refl refl refl refl = - refl -``` diff --git a/src/foundation/commuting-squares-of-maps.lagda.md b/src/foundation/commuting-squares-of-maps.lagda.md index aeaece94f2..cb2b53f6f1 100644 --- a/src/foundation/commuting-squares-of-maps.lagda.md +++ b/src/foundation/commuting-squares-of-maps.lagda.md @@ -9,26 +9,25 @@ open import foundation-core.commuting-squares-of-maps public
Imports ```agda -open import foundation.action-on-higher-identifications-functions open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions -open import foundation.commuting-squares-of-homotopies -open import foundation.commuting-squares-of-identifications open import foundation.commuting-triangles-of-maps open import foundation.function-extensionality -open import foundation.identity-types open import foundation.postcomposition-functions open import foundation.precomposition-functions open import foundation.transposition-identifications-along-equivalences open import foundation.universe-levels open import foundation.whiskering-higher-homotopies-composition open import foundation.whiskering-homotopies-composition -open import foundation.whiskering-identifications-concatenation open import foundation-core.commuting-prisms-of-maps +open import foundation-core.commuting-squares-of-homotopies +open import foundation-core.commuting-squares-of-identifications open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.homotopies +open import foundation-core.identity-types +open import foundation-core.whiskering-identifications-concatenation ```
@@ -175,7 +174,7 @@ module _ ( id) ( is-retraction-map-inv-equiv left) ( H) - ( coherence-square-maps-inv-equiv-vertical top left right bottom H) + ( vertical-inv-equiv-coherence-square-maps top left right bottom H) ( refl-htpy) ( is-retraction-map-inv-equiv right) left-inverse-law-pasting-vertical-coherence-square-maps H a = @@ -284,7 +283,7 @@ module _ ( id) ( id) ( is-section-map-inv-equiv left) - ( coherence-square-maps-inv-equiv-vertical top left right bottom H) + ( vertical-inv-equiv-coherence-square-maps top left right bottom H) ( H) ( refl-htpy) ( is-section-map-inv-equiv right) @@ -295,7 +294,7 @@ module _ ( H (map-inv-equiv left a)) ( ap ( map-equiv right) - ( coherence-square-maps-inv-equiv-vertical top left right bottom + ( vertical-inv-equiv-coherence-square-maps top left right bottom ( H) ( a))) ( is-section-map-inv-equiv right (bottom a))) ∙ @@ -570,8 +569,12 @@ module _ ( sq-left-bottom ·r left-top) ( mid-bottom ·l sq-left-top)) ∙h ( double-whisker-coherence-square-homotopies - ( bottom-right ·l (sq-left-bottom ·r left-top)) - ( right-bottom ·l (sq-right-top ·r top-left)) + ( bottom-right ·l sq-left-bottom ·r left-top) + ( sq-right-bottom ·r mid-left ·r left-top) + ( bottom-right ·l mid-bottom ·l sq-left-top) + ( right-bottom ·l mid-right ·l sq-left-top) + ( sq-right-bottom ·r mid-top ·r top-left) + ( right-bottom ·l sq-right-top ·r top-left) ( inv-htpy ( swap-nat-coherence-square-maps ( top-left) diff --git a/src/foundation/commuting-triangles-of-homotopies.lagda.md b/src/foundation/commuting-triangles-of-homotopies.lagda.md index 381b508986..d3f3b61b72 100644 --- a/src/foundation/commuting-triangles-of-homotopies.lagda.md +++ b/src/foundation/commuting-triangles-of-homotopies.lagda.md @@ -11,11 +11,11 @@ open import foundation.action-on-identifications-functions open import foundation.commuting-triangles-of-identifications open import foundation.universe-levels open import foundation.whiskering-homotopies-composition -open import foundation.whiskering-identifications-concatenation open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types +open import foundation-core.whiskering-identifications-concatenation ``` diff --git a/src/foundation/commuting-triangles-of-identifications.lagda.md b/src/foundation/commuting-triangles-of-identifications.lagda.md index 27301a5608..071dbaa96f 100644 --- a/src/foundation/commuting-triangles-of-identifications.lagda.md +++ b/src/foundation/commuting-triangles-of-identifications.lagda.md @@ -137,7 +137,7 @@ transformations are equivalences. These operations are useful in proofs involving [path algebra](foundation.path-algebra.md), because taking -`equiv-right-whisker-triangle-identicications` as an example, it provides us +`equiv-right-whisker-triangle-identifications` as an example, it provides us with two maps: the forward direction states `(p ＝ q ∙ r) → (p ∙ s ＝ q ∙ (r ∙ s))`, which allows one to append an identification without needing to reassociate on the right, and the backwards diff --git a/src/foundation/commuting-triangles-of-maps.lagda.md b/src/foundation/commuting-triangles-of-maps.lagda.md index d4163b847b..742618605e 100644 --- a/src/foundation/commuting-triangles-of-maps.lagda.md +++ b/src/foundation/commuting-triangles-of-maps.lagda.md @@ -17,11 +17,11 @@ open import foundation.identity-types open import foundation.postcomposition-functions open import foundation.precomposition-functions open import foundation.universe-levels -open import foundation.whiskering-identifications-concatenation open import foundation-core.commuting-squares-of-maps open import foundation-core.equivalences open import foundation-core.function-types +open import foundation-core.whiskering-identifications-concatenation ``` @@ -196,7 +196,7 @@ module _ coherence-htpy-triangle-maps : left ~ left' → right ~ right' → top ~ top' → UU (l1 ⊔ l2) coherence-htpy-triangle-maps L R T = - c ∙h horizontal-concat-htpy T R ~ L ∙h c' + c ∙h horizontal-concat-htpy R T ~ L ∙h c' ``` ### Pasting commuting triangles into commuting squares along homotopic diagonals diff --git a/src/foundation/embeddings.lagda.md b/src/foundation/embeddings.lagda.md index 85b638587e..31ac2aad1d 100644 --- a/src/foundation/embeddings.lagda.md +++ b/src/foundation/embeddings.lagda.md @@ -10,7 +10,6 @@ open import foundation-core.embeddings public ```agda open import foundation.action-on-identifications-functions -open import foundation.commuting-squares-of-maps open import foundation.cones-over-cospan-diagrams open import foundation.dependent-pair-types open import foundation.equivalences @@ -23,6 +22,7 @@ open import foundation.truncated-maps open import foundation.universe-levels open import foundation-core.cartesian-product-types +open import foundation-core.commuting-squares-of-maps open import foundation-core.commuting-triangles-of-maps open import foundation-core.contractible-types open import foundation-core.fibers-of-maps @@ -407,7 +407,7 @@ module _ ( map-inv-is-equiv K) ( map-inv-is-equiv L) ( top) - ( coherence-square-maps-inv-equiv-vertical + ( vertical-inv-equiv-coherence-square-maps ( top) ( left , K) ( right , L) diff --git a/src/foundation/equivalences-arrows.lagda.md b/src/foundation/equivalences-arrows.lagda.md index 98c901b158..eac852d2c4 100644 --- a/src/foundation/equivalences-arrows.lagda.md +++ b/src/foundation/equivalences-arrows.lagda.md @@ -218,7 +218,7 @@ module _ ( equiv-domain-inv-equiv-arrow) ( equiv-codomain-inv-equiv-arrow) coh-inv-equiv-arrow = - coherence-square-maps-inv-equiv-horizontal + horizontal-inv-equiv-coherence-square-maps ( equiv-domain-equiv-arrow f g α) ( f) ( g) diff --git a/src/foundation/equivalences.lagda.md b/src/foundation/equivalences.lagda.md index 7bdc8ba6a1..9ebb42a77a 100644 --- a/src/foundation/equivalences.lagda.md +++ b/src/foundation/equivalences.lagda.md @@ -15,13 +15,10 @@ open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.function-extensionality open import foundation.functoriality-fibers-of-maps -open import foundation.identity-types -open import foundation.path-algebra open import foundation.transposition-identifications-along-equivalences open import foundation.truncated-maps open import foundation.universal-property-equivalences open import foundation.universe-levels -open import foundation.whiskering-identifications-concatenation open import foundation-core.commuting-triangles-of-maps open import foundation-core.contractible-maps @@ -31,6 +28,7 @@ open import foundation-core.fibers-of-maps open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies +open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.pullbacks @@ -604,8 +602,8 @@ module _ is-pullback-swap-cone' f g c ( is-pullback-is-equiv-vertical-maps g f ( swap-cone f g c) - is-equiv-f - is-equiv-q) + ( is-equiv-f) + ( is-equiv-q)) ``` ## See also diff --git a/src/foundation/fundamental-theorem-of-identity-types.lagda.md b/src/foundation/fundamental-theorem-of-identity-types.lagda.md index 9614810977..a82990327e 100644 --- a/src/foundation/fundamental-theorem-of-identity-types.lagda.md +++ b/src/foundation/fundamental-theorem-of-identity-types.lagda.md @@ -145,7 +145,7 @@ module _ ((x : A) → section (f x)) → is-fiberwise-equiv f fundamental-theorem-id-section f section-f x = - is-equiv-section-is-equiv + is-equiv-is-equiv-section ( f x) ( section-f x) ( fundamental-theorem-id-retraction diff --git a/src/foundation/homotopies.lagda.md b/src/foundation/homotopies.lagda.md index f047aafe53..124dac3bbf 100644 --- a/src/foundation/homotopies.lagda.md +++ b/src/foundation/homotopies.lagda.md @@ -13,7 +13,6 @@ open import foundation.action-on-higher-identifications-functions open import foundation.action-on-identifications-dependent-functions open import foundation.action-on-identifications-functions open import foundation.binary-equivalences -open import foundation.commuting-squares-of-identifications open import foundation.dependent-pair-types open import foundation.function-extensionality open import foundation.homotopy-induction @@ -21,13 +20,14 @@ open import foundation.identity-types open import foundation.path-algebra open import foundation.universe-levels open import foundation.whiskering-homotopies-composition -open import foundation.whiskering-identifications-concatenation +open import foundation-core.commuting-squares-of-identifications open import foundation-core.dependent-identifications open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.transport-along-identifications +open import foundation-core.whiskering-identifications-concatenation ``` diff --git a/src/foundation/homotopy-algebra.lagda.md b/src/foundation/homotopy-algebra.lagda.md index 43a59baced..80e83db492 100644 --- a/src/foundation/homotopy-algebra.lagda.md +++ b/src/foundation/homotopy-algebra.lagda.md @@ -7,7 +7,6 @@ module foundation.homotopy-algebra where
Imports ```agda -open import foundation.homotopy-induction open import foundation.universe-levels open import foundation.whiskering-homotopies-composition @@ -34,12 +33,12 @@ module _ {f f' : (x : A) → B x} {g g' : {x : A} → B x → C x} where - horizontal-concat-htpy : f ~ f' → ({x : A} → g {x} ~ g' {x}) → g ∘ f ~ g' ∘ f' - horizontal-concat-htpy F G = (g ·l F) ∙h (G ·r f') + horizontal-concat-htpy : ({x : A} → g {x} ~ g' {x}) → f ~ f' → g ∘ f ~ g' ∘ f' + horizontal-concat-htpy G F = (g ·l F) ∙h (G ·r f') horizontal-concat-htpy' : - f ~ f' → ({x : A} → g {x} ~ g' {x}) → g ∘ f ~ g' ∘ f' - horizontal-concat-htpy' F G = (G ·r f) ∙h (g' ·l F) + ({x : A} → g {x} ~ g' {x}) → f ~ f' → g ∘ f ~ g' ∘ f' + horizontal-concat-htpy' G F = (G ·r f) ∙h (g' ·l F) ``` ## Properties @@ -51,27 +50,43 @@ module _ {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} where - coh-left-unit-horizontal-concat-htpy : + coh-right-unit-horizontal-concat-htpy : {f : (x : A) → B x} {g g' : {x : A} → B x → C x} (G : {x : A} → g {x} ~ g' {x}) → - horizontal-concat-htpy (refl-htpy' f) G ~ - horizontal-concat-htpy' (refl-htpy' f) G - coh-left-unit-horizontal-concat-htpy G = inv-htpy-right-unit-htpy + horizontal-concat-htpy G (refl-htpy' f) ~ + horizontal-concat-htpy' G (refl-htpy' f) + coh-right-unit-horizontal-concat-htpy G = inv-htpy-right-unit-htpy - coh-right-unit-horizontal-concat-htpy : + coh-left-unit-horizontal-concat-htpy : {f f' : (x : A) → B x} {g : {x : A} → B x → C x} (F : f ~ f') → - horizontal-concat-htpy F (refl-htpy' g) ~ - horizontal-concat-htpy' F (refl-htpy' g) - coh-right-unit-horizontal-concat-htpy F = right-unit-htpy + horizontal-concat-htpy (refl-htpy' g) F ~ + horizontal-concat-htpy' (refl-htpy' g) F + coh-left-unit-horizontal-concat-htpy F = right-unit-htpy +``` + +For the general case, we must construct a coherence of the square + +```text + g ·r F + gf -------> gf' + | | + G ·r f | | G ·r f' + ∨ ∨ + g'f ------> g'f' + g' ·r F +``` + +but this is an instance of naturality of `G` applied to `F`. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + {f f' : (x : A) → B x} {g g' : {x : A} → B x → C x} + (G : {x : A} → g {x} ~ g' {x}) (F : f ~ f') + where coh-horizontal-concat-htpy : - {f f' : (x : A) → B x} {g g' : {x : A} → B x → C x} - (F : f ~ f') (G : {x : A} → g {x} ~ g' {x}) → - horizontal-concat-htpy F G ~ horizontal-concat-htpy' F G - coh-horizontal-concat-htpy {f} F G = - ind-htpy f - ( λ f' F → horizontal-concat-htpy F G ~ horizontal-concat-htpy' F G) - ( coh-left-unit-horizontal-concat-htpy G) - ( F) + horizontal-concat-htpy' G F ~ horizontal-concat-htpy G F + coh-horizontal-concat-htpy = nat-htpy G ·r F ``` diff --git a/src/foundation/invertible-maps.lagda.md b/src/foundation/invertible-maps.lagda.md index 6f2b2ec088..47fb7cca31 100644 --- a/src/foundation/invertible-maps.lagda.md +++ b/src/foundation/invertible-maps.lagda.md @@ -9,6 +9,7 @@ open import foundation-core.invertible-maps public
Imports ```agda +open import foundation.action-on-identifications-functions open import foundation.commuting-triangles-of-homotopies open import foundation.dependent-pair-types open import foundation.equality-cartesian-product-types @@ -22,7 +23,6 @@ open import foundation.homotopies open import foundation.homotopy-algebra open import foundation.homotopy-induction open import foundation.postcomposition-functions -open import foundation.propositions open import foundation.retractions open import foundation.sections open import foundation.structure-identity-principle @@ -31,9 +31,11 @@ open import foundation.universe-levels open import foundation.whiskering-homotopies-composition open import foundation-core.cartesian-product-types +open import foundation-core.coherently-invertible-maps open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types +open import foundation-core.propositions open import foundation-core.torsorial-type-families open import foundation-core.truncated-types open import foundation-core.truncation-levels @@ -110,9 +112,9 @@ module _ ( is-torsorial-htpy (map-inv-is-invertible s)) ( map-inv-is-invertible s , refl-htpy) ( is-torsorial-Eq-structure - ( is-torsorial-htpy (is-retraction-is-invertible s)) - ( is-retraction-is-invertible s , refl-htpy) - (is-torsorial-htpy (is-section-is-invertible s))) + ( is-torsorial-htpy (is-section-map-inv-is-invertible s)) + ( is-section-map-inv-is-invertible s , refl-htpy) + (is-torsorial-htpy (is-retraction-map-inv-is-invertible s))) is-equiv-htpy-eq-is-invertible : (s t : is-invertible f) → is-equiv (htpy-eq-is-invertible s t) @@ -144,13 +146,13 @@ module _ map-inv-invertible-map s ~ map-inv-invertible-map t → UU (l1 ⊔ l2) coherence-htpy-invertible-map s t H I = ( coherence-triangle-homotopies - ( is-retraction-map-invertible-map s) - ( is-retraction-map-invertible-map t) - ( horizontal-concat-htpy I H)) × + ( is-section-map-inv-invertible-map s) + ( is-section-map-inv-invertible-map t) + ( horizontal-concat-htpy H I)) × ( coherence-triangle-homotopies - ( is-section-map-invertible-map s) - ( is-section-map-invertible-map t) - ( horizontal-concat-htpy H I)) + ( is-retraction-map-inv-invertible-map s) + ( is-retraction-map-inv-invertible-map t) + ( horizontal-concat-htpy I H)) htpy-invertible-map : (s t : invertible-map A B) → UU (l1 ⊔ l2) htpy-invertible-map s t = @@ -179,9 +181,9 @@ module _ ( is-torsorial-htpy (map-inv-invertible-map s)) ( map-inv-invertible-map s , refl-htpy) ( is-torsorial-Eq-structure - ( is-torsorial-htpy (is-retraction-map-invertible-map s)) - ( is-retraction-map-invertible-map s , refl-htpy) - ( is-torsorial-htpy (is-section-map-invertible-map s)))) + ( is-torsorial-htpy (is-section-map-inv-invertible-map s)) + ( is-section-map-inv-invertible-map s , refl-htpy) + ( is-torsorial-htpy (is-retraction-map-inv-invertible-map s)))) is-equiv-htpy-eq-invertible-map : (s t : invertible-map A B) → is-equiv (htpy-eq-invertible-map s t) @@ -375,6 +377,20 @@ module _ equiv-free-loop-equivalence-invertible-equivalence ``` +### The action of invertible maps on identifications is invertible + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + (H : is-invertible f) {x y : A} + where + + is-invertible-ap-is-invertible : is-invertible (ap f {x} {y}) + is-invertible-ap-is-invertible = + is-invertible-ap-is-coherently-invertible + ( is-coherently-invertible-is-invertible H) +``` + ## See also - For the coherent notion of invertible maps see diff --git a/src/foundation/path-algebra.lagda.md b/src/foundation/path-algebra.lagda.md index bb110a0312..acd69ff4b6 100644 --- a/src/foundation/path-algebra.lagda.md +++ b/src/foundation/path-algebra.lagda.md @@ -11,16 +11,16 @@ open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.binary-embeddings open import foundation.binary-equivalences -open import foundation.commuting-squares-of-identifications open import foundation.dependent-pair-types open import foundation.identity-types open import foundation.universe-levels -open import foundation.whiskering-identifications-concatenation +open import foundation-core.commuting-squares-of-identifications open import foundation-core.constant-maps open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.homotopies +open import foundation-core.whiskering-identifications-concatenation ```
diff --git a/src/foundation/postcomposition-dependent-functions.lagda.md b/src/foundation/postcomposition-dependent-functions.lagda.md index 137f0f6cca..7e35bf5c87 100644 --- a/src/foundation/postcomposition-dependent-functions.lagda.md +++ b/src/foundation/postcomposition-dependent-functions.lagda.md @@ -102,7 +102,7 @@ module _ ( eq-htpy) ( ap (postcomp-Π A f)) compute-eq-htpy-ap-postcomp-Π = - coherence-square-maps-inv-equiv-vertical + vertical-inv-equiv-coherence-square-maps ( ap (postcomp-Π A f)) ( equiv-funext) ( equiv-funext) diff --git a/src/foundation/precomposition-dependent-functions.lagda.md b/src/foundation/precomposition-dependent-functions.lagda.md index 9ef0b0863b..a4ef43be00 100644 --- a/src/foundation/precomposition-dependent-functions.lagda.md +++ b/src/foundation/precomposition-dependent-functions.lagda.md @@ -91,7 +91,7 @@ module _ ( eq-htpy) ( ap (precomp-Π f C) {g} {h}) compute-eq-htpy-ap-precomp-Π = - coherence-square-maps-inv-equiv-vertical + vertical-inv-equiv-coherence-square-maps ( ap (precomp-Π f C)) ( equiv-funext) ( equiv-funext) diff --git a/src/foundation/precomposition-functions.lagda.md b/src/foundation/precomposition-functions.lagda.md index c667f7f2f3..3f21781779 100644 --- a/src/foundation/precomposition-functions.lagda.md +++ b/src/foundation/precomposition-functions.lagda.md @@ -206,7 +206,7 @@ module _ ( eq-htpy) ( ap (precomp f C)) compute-eq-htpy-ap-precomp = - coherence-square-maps-inv-equiv-vertical + vertical-inv-equiv-coherence-square-maps ( ap (precomp f C)) ( equiv-funext) ( equiv-funext) diff --git a/src/foundation/retractions.lagda.md b/src/foundation/retractions.lagda.md index a58a62b32d..3c51b23ceb 100644 --- a/src/foundation/retractions.lagda.md +++ b/src/foundation/retractions.lagda.md @@ -12,6 +12,7 @@ open import foundation-core.retractions public open import foundation.action-on-identifications-functions open import foundation.coslice open import foundation.dependent-pair-types +open import foundation.function-extensionality open import foundation.retracts-of-types open import foundation.universe-levels open import foundation.whiskering-homotopies-composition @@ -97,25 +98,64 @@ pr2 ```agda abstract is-injective-retraction : - {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → retraction f → - is-injective f - is-injective-retraction f (h , H) {x} {y} p = (inv (H x)) ∙ (ap h p ∙ H y) + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → + retraction f → is-injective f + is-injective-retraction f (h , H) {x} {y} p = inv (H x) ∙ (ap h p ∙ H y) ``` ### Transposing identifications along retractions ```agda module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) (g : B → A) where - transpose-eq-retraction : - (g : B → A) (H : (g ∘ f) ~ id) {x : B} {y : A} → - x ＝ f y → g x ＝ y - transpose-eq-retraction g H refl = H _ - - transpose-eq-retraction' : - (g : B → A) (H : (g ∘ f) ~ id) {x : A} {y : B} → - f x ＝ y → x ＝ g y - transpose-eq-retraction' g H refl = inv (H _) + transpose-eq-is-retraction : + g ∘ f ~ id → {x : B} {y : A} → x ＝ f y → g x ＝ y + transpose-eq-is-retraction H {x} {y} p = ap g p ∙ H y + + transpose-eq-is-retraction' : + g ∘ f ~ id → {x : A} {y : B} → f x ＝ y → x ＝ g y + transpose-eq-is-retraction' H {x} refl = inv (H x) + + is-retraction-transpose-eq : + ({x : B} {y : A} → x ＝ f y → g x ＝ y) → g ∘ f ~ id + is-retraction-transpose-eq H x = H refl + + is-retraction-transpose-eq' : + ({x : A} {y : B} → f x ＝ y → x ＝ g y) → g ∘ f ~ id + is-retraction-transpose-eq' H x = inv (H refl) + + is-retraction-is-retraction-transpose-eq : + is-retraction-transpose-eq ∘ transpose-eq-is-retraction ~ id + is-retraction-is-retraction-transpose-eq H = refl + + htpy-is-section-is-retraction-transpose-eq : + (H : {x : B} {y : A} → x ＝ f y → g x ＝ y) + (x : B) (y : A) → + transpose-eq-is-retraction (is-retraction-transpose-eq H) {x} {y} ~ + H {x} {y} + htpy-is-section-is-retraction-transpose-eq H x y refl = refl + + abstract + is-section-is-retraction-transpose-eq : + transpose-eq-is-retraction ∘ is-retraction-transpose-eq ~ id + is-section-is-retraction-transpose-eq H = + eq-htpy-implicit + ( λ x → + eq-htpy-implicit + ( λ y → eq-htpy (htpy-is-section-is-retraction-transpose-eq H x y))) + + is-equiv-transpose-eq-is-retraction : + is-equiv transpose-eq-is-retraction + is-equiv-transpose-eq-is-retraction = + is-equiv-is-invertible + ( is-retraction-transpose-eq) + ( is-section-is-retraction-transpose-eq) + ( is-retraction-is-retraction-transpose-eq) + + equiv-transpose-eq-is-retraction : + (g ∘ f ~ id) ≃ ({x : B} {y : A} → x ＝ f y → g x ＝ y) + equiv-transpose-eq-is-retraction = + (transpose-eq-is-retraction , is-equiv-transpose-eq-is-retraction) ``` diff --git a/src/foundation/sections.lagda.md b/src/foundation/sections.lagda.md index 15ba000b59..ca2ce02401 100644 --- a/src/foundation/sections.lagda.md +++ b/src/foundation/sections.lagda.md @@ -192,16 +192,55 @@ is-injective-map-section-family b = ap pr1 ```agda module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) (g : B → A) where - transpose-eq-section : - (g : B → A) (H : (f ∘ g) ~ id) {x : A} {y : B} → - x ＝ g y → f x ＝ y - transpose-eq-section g H refl = H _ - - transpose-eq-section' : - (g : B → A) (H : (f ∘ g) ~ id) {x : B} {y : A} → - g x ＝ y → x ＝ f y - transpose-eq-section' g H refl = inv (H _) + transpose-eq-is-section : + f ∘ g ~ id → {x : A} {y : B} → x ＝ g y → f x ＝ y + transpose-eq-is-section H {x} {y} p = ap f p ∙ H y + + transpose-eq-is-section' : + f ∘ g ~ id → {x : B} {y : A} → g x ＝ y → x ＝ f y + transpose-eq-is-section' H {x} refl = inv (H x) + + is-section-transpose-eq : + ({x : A} {y : B} → x ＝ g y → f x ＝ y) → f ∘ g ~ id + is-section-transpose-eq H x = H refl + + is-section-transpose-eq' : + ({x : B} {y : A} → g x ＝ y → x ＝ f y) → f ∘ g ~ id + is-section-transpose-eq' H x = inv (H refl) + + is-retraction-is-section-transpose-eq : + is-section-transpose-eq ∘ transpose-eq-is-section ~ id + is-retraction-is-section-transpose-eq H = refl + + htpy-is-section-is-section-transpose-eq : + (H : {x : A} {y : B} → x ＝ g y → f x ＝ y) → + (x : A) (y : B) → + transpose-eq-is-section (is-section-transpose-eq H) {x} {y} ~ H {x} {y} + htpy-is-section-is-section-transpose-eq H x y refl = refl + + abstract + is-section-is-section-transpose-eq : + transpose-eq-is-section ∘ is-section-transpose-eq ~ id + is-section-is-section-transpose-eq H = + eq-htpy-implicit + ( λ x → + eq-htpy-implicit + ( λ y → + eq-htpy (htpy-is-section-is-section-transpose-eq H x y))) + + is-equiv-transpose-eq-is-section : + is-equiv transpose-eq-is-section + is-equiv-transpose-eq-is-section = + is-equiv-is-invertible + ( is-section-transpose-eq) + ( is-section-is-section-transpose-eq) + ( is-retraction-is-section-transpose-eq) + + equiv-transpose-eq-is-section : + (f ∘ g ~ id) ≃ ({x : A} {y : B} → x ＝ g y → f x ＝ y) + equiv-transpose-eq-is-section = + (transpose-eq-is-section , is-equiv-transpose-eq-is-section) ``` diff --git a/src/foundation/whiskering-higher-homotopies-composition.lagda.md b/src/foundation/whiskering-higher-homotopies-composition.lagda.md index eb981d65af..2970a34a1e 100644 --- a/src/foundation/whiskering-higher-homotopies-composition.lagda.md +++ b/src/foundation/whiskering-higher-homotopies-composition.lagda.md @@ -38,8 +38,7 @@ by ``` Similarly, the {{#concept "right whiskering" Disambiguation="2-homotopies with -respect to composition" Agda=right-whisker-comp²]} is defined to be the -operation +respect to composition" Agda=right-whisker-comp²}} is defined to be the operation ```text (H ~ H') → (h : (x : A) → B x) → (H ·r h ~ H' ·r h) @@ -81,3 +80,20 @@ module _ (α : {x : A} → H {x} ~ H' {x}) (h : (x : A) → B x) → H ·r h ~ H' ·r h right-whisker-comp² α h = α ·r h ``` + +### Double whiskering higher homotopies + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} + {C : (x : A) → B x → UU l3} {D : (x : A) → B x → UU l4} + {f g : {x : A} (y : B x) → C x y} {H H' : {x : A} → f {x} ~ g {x}} + where + + double-whisker-comp² : + (left : {x : A} {y : B x} → C x y → D x y) + (α : {x : A} → H {x} ~ H' {x}) + (right : (x : A) → B x) → + left ·l H ·r right ~ left ·l H' ·r right + double-whisker-comp² left α right = double-whisker-comp (ap left) α right +``` diff --git a/src/foundation/whiskering-homotopies-composition.lagda.md b/src/foundation/whiskering-homotopies-composition.lagda.md index 12a6729411..2715167d25 100644 --- a/src/foundation/whiskering-homotopies-composition.lagda.md +++ b/src/foundation/whiskering-homotopies-composition.lagda.md @@ -235,11 +235,18 @@ module _ {A : UU l1} {B : A → UU l2} {C : A → UU l3} {D : A → UU l4} where + inv-preserves-comp-left-whisker-comp : + ( k : {x : A} → C x → D x) (h : {x : A} → B x → C x) {f g : (x : A) → B x} + ( H : f ~ g) → + (k ∘ h) ·l H ~ k ·l (h ·l H) + inv-preserves-comp-left-whisker-comp k h H x = ap-comp k h (H x) + preserves-comp-left-whisker-comp : ( k : {x : A} → C x → D x) (h : {x : A} → B x → C x) {f g : (x : A) → B x} ( H : f ~ g) → k ·l (h ·l H) ~ (k ∘ h) ·l H - preserves-comp-left-whisker-comp k h H x = inv (ap-comp k h (H x)) + preserves-comp-left-whisker-comp k h H = + inv-htpy (inv-preserves-comp-left-whisker-comp k h H) module _ { l1 l2 l3 l4 : Level} @@ -271,8 +278,8 @@ module _ coh-htpy-id : H ·r f ~ f ·l H coh-htpy-id x = is-injective-concat' (H x) (nat-htpy-id H (H x)) - inv-htpy-coh-htpy-id : f ·l H ~ H ·r f - inv-htpy-coh-htpy-id = inv-htpy coh-htpy-id + inv-coh-htpy-id : f ·l H ~ H ·r f + inv-coh-htpy-id = inv-htpy coh-htpy-id ``` ## See also diff --git a/src/foundation/whiskering-homotopies-concatenation.lagda.md b/src/foundation/whiskering-homotopies-concatenation.lagda.md index 9803a73be7..dffb6ea28b 100644 --- a/src/foundation/whiskering-homotopies-concatenation.lagda.md +++ b/src/foundation/whiskering-homotopies-concatenation.lagda.md @@ -2,17 +2,19 @@ ```agda module foundation.whiskering-homotopies-concatenation where + +open import foundation-core.whiskering-homotopies-concatenation public ```
Imports ```agda -open import foundation.functoriality-dependent-function-types open import foundation.universe-levels open import foundation.whiskering-identifications-concatenation open import foundation.whiskering-operations open import foundation-core.equivalences +open import foundation-core.functoriality-dependent-function-types open import foundation-core.homotopies ``` @@ -32,57 +34,13 @@ of homotopies with respect to concatenation is a ``` Similarly, we introduce -{{#concept "right whiskering" Disambiguation="homotopies with respect to concatenation' Agda=right-whisker-concat-htpy}} +{{#concept "right whiskering" Disambiguation="homotopies with respect to concatenation" Agda=right-whisker-concat-htpy}} to be an operation ```text {H I : f ~ g} → H ~ I → (J : g ~ h) → H ∙h J ~ I ∙h J. ``` -## Definitions - -### Left whiskering of homotopies with respect to concatenation - -Left whiskering of homotopies with respect to concatenation is an operation - -```text - (H : f ~ g) {I J : g ~ h} → I ~ J → H ∙h I ~ H ∙h K. -``` - -We implement the left whiskering operation of homotopies with respect to -concatenation as an instance of a general left whiskering operation. - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - where - - left-whisker-concat-htpy : - left-whiskering-operation ((x : A) → B x) (_~_) (_∙h_) (_~_) - left-whisker-concat-htpy H K x = left-whisker-concat (H x) (K x) -``` - -### Right whiskering of homotopies with respect to concatenation - -Right whiskering of homotopies with respect to concatenation is an operation - -```text - {H I : f ~ g} → H ~ I → (J : g ~ h) → H ∙h J ~ I ∙h J. -``` - -We implement the right whiskering operation of homotopies with respect to -concatenation as an instance of a general right whiskering operation. - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - where - - right-whisker-concat-htpy : - right-whiskering-operation ((x : A) → B x) (_~_) (_∙h_) (_~_) - right-whisker-concat-htpy K J x = right-whisker-concat (K x) (J x) -``` - ## Properties ### Left whiskering of homotopies with respect to concatenation is an equivalence @@ -114,44 +72,3 @@ module _ is-equiv-map-Π-is-fiberwise-equiv ( λ x → is-equiv-right-whisker-concat (J x)) ``` - -### The unit and absorption laws for left whiskering of homotopies with respect to concatenation - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - where - - left-unit-law-left-whisker-concat-htpy : - {f g : (x : A) → B x} {I J : f ~ g} (K : I ~ J) → - left-whisker-concat-htpy refl-htpy K ~ K - left-unit-law-left-whisker-concat-htpy K x = - left-unit-law-left-whisker-concat (K x) - - right-absorption-law-left-whisker-concat-htpy : - {f g h : (x : A) → B x} (H : f ~ g) {I : g ~ h} → - left-whisker-concat-htpy H (refl-htpy' I) ~ refl-htpy - right-absorption-law-left-whisker-concat-htpy H x = - right-absorption-law-left-whisker-concat (H x) _ -``` - -### The unit and absorption laws for right whiskering of homotopies with respect to concatenation - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - where - - left-absorption-law-right-whisker-concat-htpy : - {f g h : (x : A) → B x} {H : f ~ g} (J : g ~ h) → - right-whisker-concat-htpy (refl-htpy' H) J ~ refl-htpy - left-absorption-law-right-whisker-concat-htpy J x = - left-absorption-law-right-whisker-concat _ (J x) - - right-unit-law-right-whisker-concat-htpy : - {f g : (x : A) → B x} {I J : f ~ g} (K : I ~ J) → - right-unit-htpy ∙h K ~ - right-whisker-concat-htpy K refl-htpy ∙h right-unit-htpy - right-unit-law-right-whisker-concat-htpy K x = - right-unit-law-right-whisker-concat (K x) -``` diff --git a/src/foundation/whiskering-identifications-concatenation.lagda.md b/src/foundation/whiskering-identifications-concatenation.lagda.md index 661b2b9c4e..c2e632249a 100644 --- a/src/foundation/whiskering-identifications-concatenation.lagda.md +++ b/src/foundation/whiskering-identifications-concatenation.lagda.md @@ -2,6 +2,8 @@ ```agda module foundation.whiskering-identifications-concatenation where + +open import foundation-core.whiskering-identifications-concatenation public ```
Imports @@ -41,62 +43,6 @@ concatenation. Since concatenation on either side is an [equivalence](foundation-core.equivalences.md), it follows that the whiskering operations are equivalences. -## Definitions - -### Left whiskering of identifications - -Left whiskering of identifications with respect to concatenation is an operation - -```text - (p : x ＝ y) {q r : y ＝ z} → q ＝ r → p ∙ q ＝ p ∙ r -``` - -on any type. - -```agda -module _ - {l : Level} {A : UU l} - where - - left-whisker-concat : left-whiskering-operation A (_＝_) (_∙_) (_＝_) - left-whisker-concat p β = ap (p ∙_) β -``` - -### Right whiskering of identifications - -Right whiskering of identifications with respect to concatenation is an -operation - -```text - {p q : x ＝ y} → p ＝ q → (r : y ＝ z) → p ∙ r ＝ q ∙ r -``` - -on any type. - -```agda -module _ - {l : Level} {A : UU l} - where - - right-whisker-concat : right-whiskering-operation A (_＝_) (_∙_) (_＝_) - right-whisker-concat α q = ap (_∙ q) α -``` - -### Double whiskering of identifications - -```agda -module _ - {l : Level} {A : UU l} - {a b c d : A} (p : a ＝ b) {r s : b ＝ c} (t : r ＝ s) (q : c ＝ d) - where - - double-whisker-concat : (p ∙ r) ∙ q ＝ (p ∙ s) ∙ q - double-whisker-concat = right-whisker-concat (left-whisker-concat p t) q - - double-whisker-concat' : p ∙ (r ∙ q) ＝ p ∙ (s ∙ q) - double-whisker-concat' = left-whisker-concat p (right-whisker-concat t q) -``` - ## Properties ### Left whiskering of identifications is an equivalence @@ -136,231 +82,3 @@ module _ pr2 equiv-right-whisker-concat = is-equiv-right-whisker-concat ``` - -### The unit and absorption laws for left whiskering of identifications - -```agda -module _ - {l : Level} {A : UU l} - where - - left-unit-law-left-whisker-concat : - {x y : A} {p p' : x ＝ y} (α : p ＝ p') → - left-whisker-concat refl α ＝ α - left-unit-law-left-whisker-concat refl = refl - - right-absorption-law-left-whisker-concat : - {x y z : A} (p : x ＝ y) (q : y ＝ z) → - left-whisker-concat p (refl {x = q}) ＝ refl - right-absorption-law-left-whisker-concat p q = refl -``` - -### The unit and absorption laws for right whiskering of identifications - -The right unit law for right whiskering of identifications with respect to -concatenation asserts that the square of identifications - -```text - right-whisker-concat α refl - p ∙ refl -----------------------------> p' ∙ refl - | | - right-unit | | - ∨ ∨ - p -------------------------------------> p' -``` - -commutes for any `α : p ＝ p'`. Note that this law is slightly more complicated, -since concatenating with `refl` on the right does not compute to the identity -function. - -```agda -module _ - {l : Level} {A : UU l} - where - - right-unit-law-right-whisker-concat : - {x y : A} {p p' : x ＝ y} (α : p ＝ p') → - right-unit ∙ α ＝ right-whisker-concat α refl ∙ right-unit - right-unit-law-right-whisker-concat {p = refl} refl = refl - - left-absorption-law-right-whisker-concat : - {x y z : A} (p : x ＝ y) (q : y ＝ z) → - right-whisker-concat (refl {x = p}) q ＝ refl - left-absorption-law-right-whisker-concat p q = refl -``` - -### Commutativity of left and right whiskering of identifications - -Consider four identifications `p p' : x ＝ y` and `q q' : y ＝ z` in a type `A`. -Then the square of identifications - -```text - right-whisker α q - p ∙ q ---------------------> p' ∙ q - | | - left-whisker p β | | left-whisker p' β - ∨ ∨ - p ∙ q' --------------------> p' ∙ q' - right-whisker α q' -``` - -commutes. There are at least two natural ways in which this square is seen to -commute: - -1. The square commutes by naturality of the homotopy - `α ↦ left-whisker-concat α β`. -2. The transposed square commutes by the naturality of the homotopy - `β ↦ right-whisker-concat α β`. - -These two ways in which the square commutes are inverse to each other. - -**Note.** The following statements could have been formalized using -[commuting squares of identifications](foundation.commuting-squares-of-identifications.md). -However, in order to avoid cyclic module dependencies in the library we avoid -doing so. - -```agda -module _ - {l : Level} {A : UU l} {x y z : A} - where - - commutative-left-whisker-right-whisker-concat : - {q q' : y ＝ z} (β : q ＝ q') {p p' : x ＝ y} (α : p ＝ p') → - left-whisker-concat p β ∙ right-whisker-concat α q' ＝ - right-whisker-concat α q ∙ left-whisker-concat p' β - commutative-left-whisker-right-whisker-concat β = - nat-htpy (λ α → left-whisker-concat α β) - - commutative-right-whisker-left-whisker-concat : - {p p' : x ＝ y} (α : p ＝ p') {q q' : y ＝ z} (β : q ＝ q') → - right-whisker-concat α q ∙ left-whisker-concat p' β ＝ - left-whisker-concat p β ∙ right-whisker-concat α q' - commutative-right-whisker-left-whisker-concat α = - nat-htpy (right-whisker-concat α) - - compute-inv-commutative-left-whisker-right-whisker-concat : - {q q' : y ＝ z} (β : q ＝ q') {p p' : x ＝ y} (α : p ＝ p') → - inv (commutative-right-whisker-left-whisker-concat α β) ＝ - commutative-left-whisker-right-whisker-concat β α - compute-inv-commutative-left-whisker-right-whisker-concat refl refl = - refl -``` - -### Swapping the order of left and right whiskering of identifications - -Consider a diagram of identifications - -```text - r - p -----> q - a -----> b -----> c -----> - s -``` - -with `t : r ＝ s`. Then the square of identifications - -```text - assoc p r q - (p ∙ r) ∙ q -------------> p ∙ (r ∙ q) - | | - double-whisker p t q | | double-whisker' p t q - ∨ ∨ - (p ∙ s) ∙ q -------------> p ∙ (s ∙ q) - assoc p s q -``` - -commutes. - -```agda -module _ - {l1 : Level} {A : UU l1} - where - - swap-double-whisker-concat : - {a b c d : A} (p : a ＝ b) {r s : b ＝ c} (t : r ＝ s) (q : c ＝ d) → - double-whisker-concat p t q ∙ assoc p s q ＝ - assoc p r q ∙ double-whisker-concat' p t q - swap-double-whisker-concat refl refl refl = refl -``` - -### The action on identifications of concatenating by `refl` on the right - -Consider an identification `r : p ＝ q` between two identifications -`p q : x ＝ y` in a type `A`. Then the square of identifications - -```text - right-whisker r refl - p ∙ refl ----------------------> q ∙ refl - | | - right-unit | | right-unit - ∨ ∨ - p -----------------------------> q - r -``` - -commutes. - -```agda -module _ - {l : Level} {A : UU l} {x y : A} {p q : x ＝ y} - where - - compute-refl-right-whisker-concat : - (r : p ＝ q) → - right-unit ∙ r ＝ right-whisker-concat r refl ∙ right-unit - compute-refl-right-whisker-concat refl = right-unit -``` - -### Left whiskering of identifications distributes over concatenation - -```agda -module _ - {l : Level} {A : UU l} - where - - distributive-left-whisker-concat-concat : - {a b c : A} (p : a ＝ b) {q r s : b ＝ c} (α : q ＝ r) (β : r ＝ s) → - left-whisker-concat p (α ∙ β) ＝ - left-whisker-concat p α ∙ left-whisker-concat p β - distributive-left-whisker-concat-concat p refl β = refl -``` - -### Right whiskering of identifications distributes over concatenation - -```agda -module _ - {l : Level} {A : UU l} - where - - distributive-right-whisker-concat-concat : - {a b c : A} {p q r : a ＝ b} (α : p ＝ q) (β : q ＝ r) (s : b ＝ c) → - right-whisker-concat (α ∙ β) s ＝ - right-whisker-concat α s ∙ right-whisker-concat β s - distributive-right-whisker-concat-concat refl β s = refl -``` - -### Left whiskering of identifications commutes with inverses of identifications - -```agda -module _ - {l : Level} {A : UU l} - where - - compute-inv-left-whisker-concat : - {a b c : A} (p : a ＝ b) {q r : b ＝ c} (s : q ＝ r) → - left-whisker-concat p (inv s) ＝ inv (left-whisker-concat p s) - compute-inv-left-whisker-concat p s = ap-inv (concat p _) s -``` - -### Right whiskering of identifications commutes with inverses of identifications - -```agda -module _ - {l : Level} {A : UU l} - where - - compute-inv-right-whisker-concat : - {a b c : A} {p q : a ＝ b} (s : p ＝ q) (r : b ＝ c) → - right-whisker-concat (inv s) r ＝ inv (right-whisker-concat s r) - compute-inv-right-whisker-concat s r = ap-inv (concat' _ r) s -``` diff --git a/src/group-theory/conjugation.lagda.md b/src/group-theory/conjugation.lagda.md index e07bdb3b39..8ab1d3121c 100644 --- a/src/group-theory/conjugation.lagda.md +++ b/src/group-theory/conjugation.lagda.md @@ -333,7 +333,7 @@ module _ {x y z : type-Group G} → y ＝ conjugation-Group G (inv-Group G x) z → conjugation-Group G x y ＝ z transpose-eq-conjugation-Group {x} {y} {z} = - transpose-eq-section + transpose-eq-is-section ( conjugation-Group G x) ( conjugation-Group G (inv-Group G x)) ( is-section-conjugation-inv-Group x) @@ -342,7 +342,7 @@ module _ {x y z : type-Group G} → conjugation-Group G (inv-Group G x) y ＝ z → y ＝ conjugation-Group G x z transpose-eq-conjugation-Group' {x} {y} {z} = - transpose-eq-section' + transpose-eq-is-section' ( conjugation-Group G x) ( conjugation-Group G (inv-Group G x)) ( is-section-conjugation-inv-Group x) @@ -351,7 +351,7 @@ module _ {x y z : type-Group G} → y ＝ conjugation-Group G x z → conjugation-Group G (inv-Group G x) y ＝ z transpose-eq-conjugation-inv-Group {x} {y} {z} = - transpose-eq-retraction + transpose-eq-is-retraction ( conjugation-Group G x) ( conjugation-Group G (inv-Group G x)) ( is-retraction-conjugation-inv-Group x) @@ -360,7 +360,7 @@ module _ {x y z : type-Group G} → conjugation-Group G x y ＝ z → y ＝ conjugation-Group G (inv-Group G x) z transpose-eq-conjugation-inv-Group' {x} {y} {z} = - transpose-eq-retraction' + transpose-eq-is-retraction' ( conjugation-Group G x) ( conjugation-Group G (inv-Group G x)) ( is-retraction-conjugation-inv-Group x) diff --git a/src/group-theory/equivalences-group-actions.lagda.md b/src/group-theory/equivalences-group-actions.lagda.md index 87c300fea4..cabae428b7 100644 --- a/src/group-theory/equivalences-group-actions.lagda.md +++ b/src/group-theory/equivalences-group-actions.lagda.md @@ -199,7 +199,7 @@ module _ (e : equiv-action-Group G X Y) → preserves-action-Group G Y X (map-inv-equiv-action-Group e) preserves-action-map-inv-equiv-action-Group (e , H) g = - coherence-square-maps-inv-equiv-horizontal + horizontal-inv-equiv-coherence-square-maps ( e) ( mul-action-Group G X g) ( mul-action-Group G Y g) diff --git a/src/orthogonal-factorization-systems/factorizations-of-maps.lagda.md b/src/orthogonal-factorization-systems/factorizations-of-maps.lagda.md index 860b38a002..8dc962ca58 100644 --- a/src/orthogonal-factorization-systems/factorizations-of-maps.lagda.md +++ b/src/orthogonal-factorization-systems/factorizations-of-maps.lagda.md @@ -161,7 +161,7 @@ module _ UU (l1 ⊔ l2) coherence-htpy-factorization-through F E R L = ( is-factorization-factorization-through F) ~ - ( horizontal-concat-htpy L R ∙h is-factorization-factorization-through E) + ( horizontal-concat-htpy R L ∙h is-factorization-factorization-through E) htpy-factorization-through : (F E : factorization-through f X) → UU (l1 ⊔ l2 ⊔ l3) diff --git a/src/structured-types/equivalences-types-equipped-with-endomorphisms.lagda.md b/src/structured-types/equivalences-types-equipped-with-endomorphisms.lagda.md index d5a4a65454..3b8b45fb50 100644 --- a/src/structured-types/equivalences-types-equipped-with-endomorphisms.lagda.md +++ b/src/structured-types/equivalences-types-equipped-with-endomorphisms.lagda.md @@ -158,7 +158,7 @@ inv-equiv-Type-With-Endomorphism : pr1 (inv-equiv-Type-With-Endomorphism X Y e) = inv-equiv (equiv-equiv-Type-With-Endomorphism X Y e) pr2 (inv-equiv-Type-With-Endomorphism X Y e) = - coherence-square-maps-inv-equiv-horizontal + horizontal-inv-equiv-coherence-square-maps ( equiv-equiv-Type-With-Endomorphism X Y e) ( endomorphism-Type-With-Endomorphism X) ( endomorphism-Type-With-Endomorphism Y) diff --git a/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md b/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md index 6bf5b5fd8c..c0454fa477 100644 --- a/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md +++ b/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md @@ -100,6 +100,7 @@ module _ pr2 eq-descent-data-circle-function-type h = ( eq-htpy ( horizontal-concat-htpy + ( coherence-square-family-with-descent-data-circle B) ( h ·l inv-htpy ( coherence-square-maps-inv-equiv @@ -109,8 +110,7 @@ module _ ( family-family-with-descent-data-circle A) ( loop-free-loop l)) ( equiv-family-with-descent-data-circle A) - ( coherence-square-family-with-descent-data-circle A))) - ( coherence-square-family-with-descent-data-circle B))) ∙ + ( coherence-square-family-with-descent-data-circle A))))) ∙ ( inv ( ( tr-function-type ( family-family-with-descent-data-circle A) diff --git a/src/synthetic-homotopy-theory/equivalences-sequential-diagrams.lagda.md b/src/synthetic-homotopy-theory/equivalences-sequential-diagrams.lagda.md index d399a7affc..b82afe8957 100644 --- a/src/synthetic-homotopy-theory/equivalences-sequential-diagrams.lagda.md +++ b/src/synthetic-homotopy-theory/equivalences-sequential-diagrams.lagda.md @@ -140,7 +140,7 @@ module _ pr1 inv-equiv-sequential-diagram n = inv-equiv (equiv-equiv-sequential-diagram B e n) pr2 inv-equiv-sequential-diagram n = - coherence-square-maps-inv-equiv-vertical + vertical-inv-equiv-coherence-square-maps ( map-sequential-diagram A n) ( equiv-equiv-sequential-diagram B e n) ( equiv-equiv-sequential-diagram B e (succ-ℕ n)) diff --git a/src/synthetic-homotopy-theory/functoriality-sequential-colimits.lagda.md b/src/synthetic-homotopy-theory/functoriality-sequential-colimits.lagda.md index b03ced31dd..61cd54ef3b 100644 --- a/src/synthetic-homotopy-theory/functoriality-sequential-colimits.lagda.md +++ b/src/synthetic-homotopy-theory/functoriality-sequential-colimits.lagda.md @@ -336,11 +336,22 @@ module _ ( ( map-sequential-colimit-hom-sequential-diagram up-c c'' ( comp-hom-sequential-diagram A B C g f)) ·l ( coherence-cocone-sequential-diagram c n)) - ( _) + ( coherence-cocone-sequential-diagram + ( map-cocone-hom-sequential-diagram + ( comp-hom-sequential-diagram A B C g f) + ( c'')) + ( n)) + ( ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram + ( up-c) + ( c'') + ( comp-hom-sequential-diagram A B C g f) + ( succ-ℕ n)) ·r + ( map-sequential-diagram A n)) ( coherence-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c ( c'') ( comp-hom-sequential-diagram A B C g f) - ( n))) ∙h + ( n)) + ( _)) ∙h ( ap-concat-htpy ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c c'' ( comp-hom-sequential-diagram A B C g f)