diff --git a/TEMPLATE.lagda.md b/TEMPLATE.lagda.md index ab7065acb1..407fdace78 100644 --- a/TEMPLATE.lagda.md +++ b/TEMPLATE.lagda.md @@ -55,5 +55,4 @@ concept-subconcept = ... 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), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1308.0729)) + [arXiv:1308.0729](https://arxiv.org/abs/1308.0729)) diff --git a/src/category-theory.lagda.md b/src/category-theory.lagda.md index 3b99e5908a..246d29dd86 100644 --- a/src/category-theory.lagda.md +++ b/src/category-theory.lagda.md @@ -26,6 +26,7 @@ open import category-theory.category-of-maps-from-small-to-large-categories publ open import category-theory.commuting-squares-of-morphisms-in-large-precategories public open import category-theory.commuting-squares-of-morphisms-in-precategories public open import category-theory.composition-operations-on-binary-families-of-sets public +open import category-theory.copresheaf-categories public open import category-theory.coproducts-in-precategories public open import category-theory.dependent-products-of-categories public open import category-theory.dependent-products-of-large-categories public @@ -59,6 +60,7 @@ open import category-theory.functors-from-small-to-large-precategories public open import category-theory.functors-large-categories public open import category-theory.functors-large-precategories public open import category-theory.functors-precategories public +open import category-theory.gaunt-categories public open import category-theory.groupoids public open import category-theory.homotopies-natural-transformations-large-precategories public open import category-theory.initial-objects-large-categories public @@ -70,6 +72,7 @@ open import category-theory.isomorphisms-in-categories public open import category-theory.isomorphisms-in-large-categories public open import category-theory.isomorphisms-in-large-precategories public open import category-theory.isomorphisms-in-precategories public +open import category-theory.isomorphisms-in-subprecategories public open import category-theory.large-categories public open import category-theory.large-function-categories public open import category-theory.large-function-precategories public @@ -88,6 +91,7 @@ open import category-theory.natural-isomorphisms-maps-categories public open import category-theory.natural-isomorphisms-maps-precategories public open import category-theory.natural-numbers-object-precategories public open import category-theory.natural-transformations-functors-categories public +open import category-theory.natural-transformations-functors-from-small-to-large-categories public open import category-theory.natural-transformations-functors-from-small-to-large-precategories public open import category-theory.natural-transformations-functors-large-categories public open import category-theory.natural-transformations-functors-large-precategories public @@ -97,7 +101,9 @@ open import category-theory.natural-transformations-maps-from-small-to-large-pre open import category-theory.natural-transformations-maps-precategories public open import category-theory.nonunital-precategories public open import category-theory.one-object-precategories public +open import category-theory.opposite-categories public open import category-theory.opposite-precategories public +open import category-theory.opposite-preunivalent-categories public open import category-theory.precategories public open import category-theory.precategory-of-functors public open import category-theory.precategory-of-functors-from-small-to-large-precategories public @@ -109,10 +115,13 @@ open import category-theory.preunivalent-categories public open import category-theory.products-in-precategories public open import category-theory.products-of-precategories public open import category-theory.pullbacks-in-precategories public +open import category-theory.replete-subprecategories public open import category-theory.representable-functors-categories public open import category-theory.representable-functors-large-precategories public open import category-theory.representable-functors-precategories public open import category-theory.representing-arrow-category public +open import category-theory.rigid-objects-categories public +open import category-theory.rigid-objects-precategories public open import category-theory.sieves-in-categories public open import category-theory.simplex-category public open import category-theory.slice-precategories public diff --git a/src/category-theory/categories.lagda.md b/src/category-theory/categories.lagda.md index b668dda0c3..c0994b4ba5 100644 --- a/src/category-theory/categories.lagda.md +++ b/src/category-theory/categories.lagda.md @@ -14,11 +14,13 @@ open import category-theory.precategories open import category-theory.preunivalent-categories open import foundation.1-types +open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.identity-types open import foundation.propositions open import foundation.sets +open import foundation.surjective-maps open import foundation.universe-levels ``` @@ -56,6 +58,10 @@ module _ is-category-Precategory : UU (l1 ⊔ l2) is-category-Precategory = type-Prop is-category-prop-Precategory + + is-prop-is-category-Precategory : is-prop is-category-Precategory + is-prop-is-category-Precategory = + is-prop-type-Prop is-category-prop-Precategory ``` ### The type of categories @@ -156,6 +162,20 @@ module _ is-emb-is-equiv (is-category-Category C x y) ``` +### The total hom-type of a preunivalent category + +```agda +total-hom-Category : + {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2) +total-hom-Category C = total-hom-Precategory (precategory-Category C) + +obj-total-hom-Category : + {l1 l2 : Level} (C : Category l1 l2) → + total-hom-Category C → + obj-Category C × obj-Category C +obj-total-hom-Category C = obj-total-hom-Precategory (precategory-Category C) +``` + ### Equalities induce morphisms ```agda @@ -207,3 +227,104 @@ module _ obj-1-type-Category = obj-1-type-Preunivalent-Category (preunivalent-category-Category C) ``` + +### The total hom-type of a category is a 1-type + +```agda +module _ + {l1 l2 : Level} (C : Category l1 l2) + where + + is-1-type-total-hom-Category : + is-1-type (total-hom-Category C) + is-1-type-total-hom-Category = + is-1-type-total-hom-Preunivalent-Category (preunivalent-category-Category C) + + total-hom-1-type-Category : 1-Type (l1 ⊔ l2) + total-hom-1-type-Category = + total-hom-1-type-Preunivalent-Category (preunivalent-category-Category C) +``` + +### A preunivalent category is a category if and only if `iso-eq` is surjective + +```agda +module _ + {l1 l2 : Level} (C : Precategory l1 l2) + where + + is-surjective-iso-eq-prop-Precategory : Prop (l1 ⊔ l2) + is-surjective-iso-eq-prop-Precategory = + Π-Prop + ( obj-Precategory C) + ( λ x → + Π-Prop + ( obj-Precategory C) + ( λ y → + is-surjective-Prop + ( iso-eq-Precategory C x y))) + + is-surjective-iso-eq-Precategory : UU (l1 ⊔ l2) + is-surjective-iso-eq-Precategory = + type-Prop is-surjective-iso-eq-prop-Precategory + + is-prop-is-surjective-iso-eq-Precategory : + is-prop is-surjective-iso-eq-Precategory + is-prop-is-surjective-iso-eq-Precategory = + is-prop-type-Prop is-surjective-iso-eq-prop-Precategory +``` + +```agda +module _ + {l1 l2 : Level} (C : Preunivalent-Category l1 l2) + where + + is-category-is-surjective-iso-eq-Preunivalent-Category : + is-surjective-iso-eq-Precategory (precategory-Preunivalent-Category C) → + is-category-Precategory (precategory-Preunivalent-Category C) + is-category-is-surjective-iso-eq-Preunivalent-Category + is-surjective-iso-eq-C x y = + is-equiv-is-emb-is-surjective + ( is-surjective-iso-eq-C x y) + ( is-preunivalent-Preunivalent-Category C x y) + + is-surjective-iso-eq-is-category-Preunivalent-Category : + is-category-Precategory (precategory-Preunivalent-Category C) → + is-surjective-iso-eq-Precategory (precategory-Preunivalent-Category C) + is-surjective-iso-eq-is-category-Preunivalent-Category + is-category-C x y = + is-surjective-is-equiv (is-category-C x y) + + is-equiv-is-category-is-surjective-iso-eq-Preunivalent-Category : + is-equiv is-category-is-surjective-iso-eq-Preunivalent-Category + is-equiv-is-category-is-surjective-iso-eq-Preunivalent-Category = + is-equiv-is-prop + ( is-prop-is-surjective-iso-eq-Precategory + ( precategory-Preunivalent-Category C)) + ( is-prop-is-category-Precategory (precategory-Preunivalent-Category C)) + ( is-surjective-iso-eq-is-category-Preunivalent-Category) + + is-equiv-is-surjective-iso-eq-is-category-Preunivalent-Category : + is-equiv is-surjective-iso-eq-is-category-Preunivalent-Category + is-equiv-is-surjective-iso-eq-is-category-Preunivalent-Category = + is-equiv-is-prop + ( is-prop-is-category-Precategory (precategory-Preunivalent-Category C)) + ( is-prop-is-surjective-iso-eq-Precategory + ( precategory-Preunivalent-Category C)) + ( is-category-is-surjective-iso-eq-Preunivalent-Category) + + equiv-is-category-is-surjective-iso-eq-Preunivalent-Category : + is-surjective-iso-eq-Precategory (precategory-Preunivalent-Category C) ≃ + is-category-Precategory (precategory-Preunivalent-Category C) + pr1 equiv-is-category-is-surjective-iso-eq-Preunivalent-Category = + is-category-is-surjective-iso-eq-Preunivalent-Category + pr2 equiv-is-category-is-surjective-iso-eq-Preunivalent-Category = + is-equiv-is-category-is-surjective-iso-eq-Preunivalent-Category + + equiv-is-surjective-iso-eq-is-category-Preunivalent-Category : + is-category-Precategory (precategory-Preunivalent-Category C) ≃ + is-surjective-iso-eq-Precategory (precategory-Preunivalent-Category C) + pr1 equiv-is-surjective-iso-eq-is-category-Preunivalent-Category = + is-surjective-iso-eq-is-category-Preunivalent-Category + pr2 equiv-is-surjective-iso-eq-is-category-Preunivalent-Category = + is-equiv-is-surjective-iso-eq-is-category-Preunivalent-Category +``` diff --git a/src/category-theory/copresheaf-categories.lagda.md b/src/category-theory/copresheaf-categories.lagda.md new file mode 100644 index 0000000000..66c99e5828 --- /dev/null +++ b/src/category-theory/copresheaf-categories.lagda.md @@ -0,0 +1,169 @@ +# Copresheaf categories + +```agda +module category-theory.copresheaf-categories where +``` + +
Imports + +```agda +open import category-theory.categories +open import category-theory.category-of-functors-from-small-to-large-categories +open import category-theory.functors-precategories +open import category-theory.large-categories +open import category-theory.large-precategories +open import category-theory.precategories +open import category-theory.precategory-of-functors-from-small-to-large-precategories + +open import foundation.category-of-sets +open import foundation.sets +open import foundation.universe-levels +``` + +
+ +## Idea + +Given a [precategory](category-theory.precategories.md) `C`, we can form its +**copresheaf [category](category-theory.large-categories.md)** as the +[large category of functors](category-theory.functors-from-small-to-large-precategories.md) +from `C`, into the [large category of sets](foundation.category-of-sets.md) + +```text + C → Set. +``` + +To this large category, there is an associated +[small category](category-theory.categories.md) of small copresheaves, taking +values in small [sets](foundation-core.sets.md). + +## Definitions + +### The large category of copresheaves on a precategory + +```agda +module _ + {l1 l2 : Level} (C : Precategory l1 l2) + where + + copresheaf-Large-Precategory : + Large-Precategory (λ l → l1 ⊔ l2 ⊔ lsuc l) (λ l l' → l1 ⊔ l2 ⊔ l ⊔ l') + copresheaf-Large-Precategory = + functor-large-precategory-Small-Large-Precategory C Set-Large-Precategory + + is-large-category-copresheaf-Large-Category : + is-large-category-Large-Precategory copresheaf-Large-Precategory + is-large-category-copresheaf-Large-Category = + is-large-category-functor-large-precategory-is-large-category-Small-Large-Precategory + ( C) + ( Set-Large-Precategory) + ( is-large-category-Set-Large-Precategory) + + copresheaf-Large-Category : + Large-Category (λ l → l1 ⊔ l2 ⊔ lsuc l) (λ l l' → l1 ⊔ l2 ⊔ l ⊔ l') + large-precategory-Large-Category copresheaf-Large-Category = + copresheaf-Large-Precategory + is-large-category-Large-Category copresheaf-Large-Category = + is-large-category-copresheaf-Large-Category +``` + +We record the components of the large category of copresheaves on a precategory. + +```agda + obj-copresheaf-Large-Category = + obj-Large-Precategory copresheaf-Large-Precategory + + hom-set-copresheaf-Large-Category = + hom-set-Large-Precategory copresheaf-Large-Precategory + + hom-copresheaf-Large-Category = + hom-Large-Precategory copresheaf-Large-Precategory + + comp-hom-copresheaf-Large-Category = + comp-hom-Large-Precategory copresheaf-Large-Precategory + + id-hom-copresheaf-Large-Category = + id-hom-Large-Precategory copresheaf-Large-Precategory + + associative-comp-hom-copresheaf-Large-Category = + associative-comp-hom-Large-Precategory copresheaf-Large-Precategory + + left-unit-law-comp-hom-copresheaf-Large-Category = + left-unit-law-comp-hom-Large-Precategory copresheaf-Large-Precategory + + right-unit-law-comp-hom-copresheaf-Large-Category = + right-unit-law-comp-hom-Large-Precategory copresheaf-Large-Precategory +``` + +### The category of small copresheaves on a precategory + +```agda +module _ + {l1 l2 : Level} (C : Precategory l1 l2) (l : Level) + where + + copresheaf-Precategory : Precategory (l1 ⊔ l2 ⊔ lsuc l) (l1 ⊔ l2 ⊔ l) + copresheaf-Precategory = + precategory-Large-Precategory (copresheaf-Large-Precategory C) l + + copresheaf-Category : Category (l1 ⊔ l2 ⊔ lsuc l) (l1 ⊔ l2 ⊔ l) + copresheaf-Category = category-Large-Category (copresheaf-Large-Category C) l +``` + +We also record the components of the category of small copresheaves on a +precategory. + +```agda + obj-copresheaf-Category = + obj-Precategory copresheaf-Precategory + + hom-set-copresheaf-Category = + hom-set-Precategory copresheaf-Precategory + + hom-copresheaf-Category = + hom-Precategory copresheaf-Precategory + + comp-hom-copresheaf-Category = + comp-hom-Precategory copresheaf-Precategory + + id-hom-copresheaf-Category = + id-hom-Precategory copresheaf-Precategory + + associative-comp-hom-copresheaf-Category = + associative-comp-hom-Precategory copresheaf-Precategory + + left-unit-law-comp-hom-copresheaf-Category = + left-unit-law-comp-hom-Precategory copresheaf-Precategory + + right-unit-law-comp-hom-copresheaf-Category = + right-unit-law-comp-hom-Precategory copresheaf-Precategory +``` + +### Sections of copresheaves + +As a choice of universe level must be made to talk about sections of +copresheaves, this notion coincides for the large and small category of +copresheaves. + +```agda +module _ + {l1 l2 l3 : Level} (C : Precategory l1 l2) + where + + section-copresheaf-Category : + (F : obj-copresheaf-Category C l3) (c : obj-Precategory C) → UU l3 + section-copresheaf-Category F c = + type-Set (obj-functor-Precategory C (Set-Precategory l3) F c) +``` + +## See also + +- [The Yoneda lemma](category-theory.yoneda-lemma-precategories.md) + +## External links + +- [Presheaf precategories](https://1lab.dev/Cat.Functor.Base.html#presheaf-precategories) + at 1lab +- [category of presheaves](https://ncatlab.org/nlab/show/category+of+presheaves) + at nlab +- [copresheaf](https://ncatlab.org/nlab/show/copresheaf) at nlab diff --git a/src/category-theory/coproducts-in-precategories.lagda.md b/src/category-theory/coproducts-in-precategories.lagda.md index 18c02fc8a9..ce33878ae2 100644 --- a/src/category-theory/coproducts-in-precategories.lagda.md +++ b/src/category-theory/coproducts-in-precategories.lagda.md @@ -14,6 +14,7 @@ open import foundation.cartesian-product-types open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.identity-types +open import foundation.iterated-dependent-product-types open import foundation.propositions open import foundation.unique-existence open import foundation.universe-levels @@ -121,7 +122,7 @@ module _ is-prop-is-coproduct-Precategory : is-prop (is-coproduct-Precategory C x y p l r) is-prop-is-coproduct-Precategory = - is-prop-Π³ (λ z f g → is-property-is-contr) + is-prop-iterated-Π 3 (λ z f g → is-property-is-contr) is-coproduct-prop-Precategory : Prop (l1 ⊔ l2) pr1 is-coproduct-prop-Precategory = is-coproduct-Precategory C x y p l r diff --git a/src/category-theory/equivalences-of-precategories.lagda.md b/src/category-theory/equivalences-of-precategories.lagda.md index d26d30138f..4487578e80 100644 --- a/src/category-theory/equivalences-of-precategories.lagda.md +++ b/src/category-theory/equivalences-of-precategories.lagda.md @@ -59,5 +59,5 @@ module _ ```agda equiv-Precategory : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) - equiv-Precategory = Σ (functor-Precategory C D) is-equiv-functor-Precategory + equiv-Precategory = Σ (functor-Precategory C D) (is-equiv-functor-Precategory) ``` diff --git a/src/category-theory/faithful-maps-precategories.lagda.md b/src/category-theory/faithful-maps-precategories.lagda.md index d4376bf7cd..2fcbc71d8c 100644 --- a/src/category-theory/faithful-maps-precategories.lagda.md +++ b/src/category-theory/faithful-maps-precategories.lagda.md @@ -15,6 +15,7 @@ open import foundation.embeddings open import foundation.equivalences open import foundation.function-types open import foundation.injective-maps +open import foundation.iterated-dependent-product-types open import foundation.propositions open import foundation.universe-levels ``` @@ -49,7 +50,8 @@ module _ is-prop-is-faithful-map-Precategory : is-prop is-faithful-map-Precategory is-prop-is-faithful-map-Precategory = - is-prop-Π² (λ x y → is-prop-is-emb (hom-map-Precategory C D F {x} {y})) + is-prop-iterated-Π 2 + ( λ x y → is-property-is-emb (hom-map-Precategory C D F {x} {y})) is-faithful-prop-map-Precategory : Prop (l1 ⊔ l2 ⊔ l4) pr1 is-faithful-prop-map-Precategory = is-faithful-map-Precategory @@ -110,7 +112,7 @@ module _ is-prop-is-injective-hom-map-Precategory : is-prop is-injective-hom-map-Precategory is-prop-is-injective-hom-map-Precategory = - is-prop-Π² + is-prop-iterated-Π 2 ( λ x y → is-prop-is-injective ( is-set-hom-Precategory C x y) diff --git a/src/category-theory/full-maps-precategories.lagda.md b/src/category-theory/full-maps-precategories.lagda.md index aac8e8fb73..06d6cef6ce 100644 --- a/src/category-theory/full-maps-precategories.lagda.md +++ b/src/category-theory/full-maps-precategories.lagda.md @@ -12,6 +12,7 @@ open import category-theory.precategories open import foundation.dependent-pair-types open import foundation.function-types +open import foundation.iterated-dependent-product-types open import foundation.propositions open import foundation.surjective-maps open import foundation.universe-levels @@ -45,7 +46,7 @@ module _ is-prop-is-full-map-Precategory : is-prop is-full-map-Precategory is-prop-is-full-map-Precategory = - is-prop-Π² + is-prop-iterated-Π 2 ( λ x y → is-prop-is-surjective (hom-map-Precategory C D F {x} {y})) is-full-prop-map-Precategory : Prop (l1 ⊔ l2 ⊔ l4) diff --git a/src/category-theory/full-subprecategories.lagda.md b/src/category-theory/full-subprecategories.lagda.md index 96cc1399f8..2437f2eda8 100644 --- a/src/category-theory/full-subprecategories.lagda.md +++ b/src/category-theory/full-subprecategories.lagda.md @@ -246,9 +246,7 @@ module _ is-category-precategory-is-category-Full-Subprecategory is-category-C X = fundamental-theorem-id ( is-torsorial-Eq-subtype - ( is-torsorial-iso-Category - ( C , is-category-C) - ( inclusion-subtype P X)) + ( is-torsorial-iso-Category (C , is-category-C) (inclusion-subtype P X)) ( is-prop-is-in-subtype P) ( inclusion-subtype P X) ( id-iso-Precategory C) diff --git a/src/category-theory/fully-faithful-maps-precategories.lagda.md b/src/category-theory/fully-faithful-maps-precategories.lagda.md index e2cdc1758d..2e532d64a4 100644 --- a/src/category-theory/fully-faithful-maps-precategories.lagda.md +++ b/src/category-theory/fully-faithful-maps-precategories.lagda.md @@ -15,6 +15,7 @@ open import category-theory.precategories open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-types +open import foundation.iterated-dependent-product-types open import foundation.propositions open import foundation.surjective-maps open import foundation.universe-levels @@ -50,7 +51,7 @@ module _ is-prop-is-fully-faithful-map-Precategory : is-prop is-fully-faithful-map-Precategory is-prop-is-fully-faithful-map-Precategory = - is-prop-Π² + is-prop-iterated-Π 2 ( λ x y → is-property-is-equiv (hom-map-Precategory C D F {x} {y})) is-fully-faithful-prop-map-Precategory : Prop (l1 ⊔ l2 ⊔ l4) diff --git a/src/category-theory/functors-precategories.lagda.md b/src/category-theory/functors-precategories.lagda.md index 061e652f80..1d7354c47c 100644 --- a/src/category-theory/functors-precategories.lagda.md +++ b/src/category-theory/functors-precategories.lagda.md @@ -19,6 +19,7 @@ open import foundation.equivalences open import foundation.function-types open import foundation.homotopies open import foundation.identity-types +open import foundation.iterated-dependent-product-types open import foundation.propositions open import foundation.subtypes open import foundation.type-arithmetic-dependent-pair-types @@ -161,27 +162,56 @@ pr2 (pr2 (pr2 (id-functor-Precategory C))) x = refl Any two compatible functors can be composed to a new functor. ```agda -comp-functor-Precategory : +module _ {l1 l2 l3 l4 l5 l6 : Level} - (C : Precategory l1 l2) (D : Precategory l3 l4) (E : Precategory l5 l6) → - functor-Precategory D E → functor-Precategory C D → functor-Precategory C E -pr1 (comp-functor-Precategory C D E G F) = - obj-functor-Precategory D E G ∘ obj-functor-Precategory C D F -pr1 (pr2 (comp-functor-Precategory C D E G F)) = - hom-functor-Precategory D E G ∘ hom-functor-Precategory C D F -pr1 (pr2 (pr2 (comp-functor-Precategory C D E G F))) g f = - ( ap - ( hom-functor-Precategory D E G) - ( preserves-comp-functor-Precategory C D F g f)) ∙ - ( preserves-comp-functor-Precategory D E G - ( hom-functor-Precategory C D F g) - ( hom-functor-Precategory C D F f)) -pr2 (pr2 (pr2 (comp-functor-Precategory C D E G F))) x = - ( ap - ( hom-functor-Precategory D E G) - ( preserves-id-functor-Precategory C D F x)) ∙ - ( preserves-id-functor-Precategory D E G - ( obj-functor-Precategory C D F x)) + (A : Precategory l1 l2) (B : Precategory l3 l4) (C : Precategory l5 l6) + (G : functor-Precategory B C) (F : functor-Precategory A B) + where + + obj-comp-functor-Precategory : obj-Precategory A → obj-Precategory C + obj-comp-functor-Precategory = + obj-functor-Precategory B C G ∘ obj-functor-Precategory A B F + + hom-comp-functor-Precategory : + {x y : obj-Precategory A} → + hom-Precategory A x y → + hom-Precategory C + ( obj-comp-functor-Precategory x) + ( obj-comp-functor-Precategory y) + hom-comp-functor-Precategory = + hom-functor-Precategory B C G ∘ hom-functor-Precategory A B F + + map-comp-functor-precategory : map-Precategory A C + pr1 map-comp-functor-precategory = obj-comp-functor-Precategory + pr2 map-comp-functor-precategory = hom-comp-functor-Precategory + + preserves-comp-comp-functor-Precategory : + preserves-comp-hom-map-Precategory A C map-comp-functor-precategory + preserves-comp-comp-functor-Precategory g f = + ( ap + ( hom-functor-Precategory B C G) + ( preserves-comp-functor-Precategory A B F g f)) ∙ + ( preserves-comp-functor-Precategory B C G + ( hom-functor-Precategory A B F g) + ( hom-functor-Precategory A B F f)) + + preserves-id-comp-functor-Precategory : + preserves-id-hom-map-Precategory A C map-comp-functor-precategory + preserves-id-comp-functor-Precategory x = + ( ap + ( hom-functor-Precategory B C G) + ( preserves-id-functor-Precategory A B F x)) ∙ + ( preserves-id-functor-Precategory B C G + ( obj-functor-Precategory A B F x)) + + comp-functor-Precategory : functor-Precategory A C + pr1 comp-functor-Precategory = obj-comp-functor-Precategory + pr1 (pr2 comp-functor-Precategory) = + hom-functor-Precategory B C G ∘ hom-functor-Precategory A B F + pr1 (pr2 (pr2 comp-functor-Precategory)) = + preserves-comp-comp-functor-Precategory + pr2 (pr2 (pr2 comp-functor-Precategory)) = + preserves-id-comp-functor-Precategory ``` ## Properties @@ -202,22 +232,17 @@ module _ is-prop-preserves-comp-hom-map-Precategory : is-prop (preserves-comp-hom-map-Precategory C D F) is-prop-preserves-comp-hom-map-Precategory = - is-prop-Π' - ( λ x → - is-prop-Π' - ( λ y → - is-prop-Π' - ( λ z → - is-prop-Π² - ( λ g f → - is-set-hom-Precategory D - ( obj-map-Precategory C D F x) - ( obj-map-Precategory C D F z) - ( hom-map-Precategory C D F - ( comp-hom-Precategory C g f)) - ( comp-hom-Precategory D - ( hom-map-Precategory C D F g) - ( hom-map-Precategory C D F f)))))) + is-prop-iterated-implicit-Π 3 + ( λ x y z → + is-prop-iterated-Π 2 + ( λ g f → + is-set-hom-Precategory D + ( obj-map-Precategory C D F x) + ( obj-map-Precategory C D F z) + ( hom-map-Precategory C D F (comp-hom-Precategory C g f)) + ( comp-hom-Precategory D + ( hom-map-Precategory C D F g) + ( hom-map-Precategory C D F f)))) preserves-comp-hom-prop-map-Precategory : Prop (l1 ⊔ l2 ⊔ l4) pr1 preserves-comp-hom-prop-map-Precategory = @@ -422,6 +447,126 @@ module _ ( is-iso-iso-Precategory C f) ``` +### Categorical laws for functor composition + +#### Unit laws for functor composition + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) (D : Precategory l3 l4) + (F : functor-Precategory C D) + where + + left-unit-law-comp-functor-Precategory : + comp-functor-Precategory C D D (id-functor-Precategory D) F ＝ F + left-unit-law-comp-functor-Precategory = + eq-eq-map-functor-Precategory C D _ _ refl + + right-unit-law-comp-functor-Precategory : + comp-functor-Precategory C C D F (id-functor-Precategory C) ＝ F + right-unit-law-comp-functor-Precategory = refl +``` + +#### Associativity of functor composition + +```agda +module _ + {l1 l1' l2 l2' l3 l3' l4 l4' : Level} + (A : Precategory l1 l1') + (B : Precategory l2 l2') + (C : Precategory l3 l3') + (D : Precategory l4 l4') + (F : functor-Precategory A B) + (G : functor-Precategory B C) + (H : functor-Precategory C D) + where + + associative-comp-functor-Precategory : + comp-functor-Precategory A B D (comp-functor-Precategory B C D H G) F ＝ + comp-functor-Precategory A C D H (comp-functor-Precategory A B C G F) + associative-comp-functor-Precategory = + eq-eq-map-functor-Precategory A D _ _ refl +``` + +#### MacLane pentagon for functor composition + +```text + (I(GH))F ---- I((GH)F) + / \ + / \ + ((IH)G)F I(H(GF)) + \ / + \ / + (IH)(GF) +``` + +The proof remains to be formalized. + +```text +module _ + {l1 l1' l2 l2' l3 l3' l4 l4' : Level} + (A : Precategory l1 l1') + (B : Precategory l2 l2') + (C : Precategory l3 l3') + (D : Precategory l4 l4') + (E : Precategory l4 l4') + (F : functor-Precategory A B) + (G : functor-Precategory B C) + (H : functor-Precategory C D) + (I : functor-Precategory D E) + where + + mac-lane-pentagon-comp-functor-Precategory : + coherence-pentagon-identifications + { x = + comp-functor-Precategory A B E + ( comp-functor-Precategory B D E I + ( comp-functor-Precategory B C D H G)) + ( F)} + { comp-functor-Precategory A D E I + ( comp-functor-Precategory A B D + ( comp-functor-Precategory B C D H G) + ( F))} + { comp-functor-Precategory A B E + ( comp-functor-Precategory B C E + ( comp-functor-Precategory C D E I H) + ( G)) + ( F)} + { comp-functor-Precategory A D E + ( I) + ( comp-functor-Precategory A C D + ( H) + ( comp-functor-Precategory A B C G F))} + { comp-functor-Precategory A C E + ( comp-functor-Precategory C D E I H) + ( comp-functor-Precategory A B C G F)} + ( associative-comp-functor-Precategory A B D E + ( F) (comp-functor-Precategory B C D H G) (I)) + ( ap + ( λ p → comp-functor-Precategory A B E p F) + ( inv (associative-comp-functor-Precategory B C D E G H I))) + ( ap + ( λ p → comp-functor-Precategory A D E I p) + ( associative-comp-functor-Precategory A B C D F G H)) + ( associative-comp-functor-Precategory A B C E + ( F) (G) (comp-functor-Precategory C D E I H)) + ( inv + ( associative-comp-functor-Precategory A C D E + (comp-functor-Precategory A B C G F) H I)) + mac-lane-pentagon-comp-functor-Precategory = {!!} +``` + ## See also - [The precategory of functors and natural transformations between precategories](category-theory.precategory-of-functors.md) + +## 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)) +2. Benedikt Ahrens and Krzysztof Kapulkin and Michael Shulman, _Univalent + Categories and the Rezk completion_ (2015) + ([arXiv:1303.0584](https://arxiv.org/abs/1303.0584), + [DOI:10.1017/S0960129514000486](https://doi.org/10.1017/S0960129514000486)) diff --git a/src/category-theory/gaunt-categories.lagda.md b/src/category-theory/gaunt-categories.lagda.md new file mode 100644 index 0000000000..d902799dff --- /dev/null +++ b/src/category-theory/gaunt-categories.lagda.md @@ -0,0 +1,359 @@ +# Gaunt categories + +```agda +module category-theory.gaunt-categories where +``` + +
Imports + +```agda +open import category-theory.categories +open import category-theory.composition-operations-on-binary-families-of-sets +open import category-theory.isomorphism-induction-categories +open import category-theory.isomorphisms-in-categories +open import category-theory.isomorphisms-in-precategories +open import category-theory.nonunital-precategories +open import category-theory.precategories +open import category-theory.preunivalent-categories +open import category-theory.rigid-objects-categories +open import category-theory.strict-categories + +open import foundation.cartesian-product-types +open import foundation.contractible-types +open import foundation.dependent-pair-types +open import foundation.identity-types +open import foundation.propositions +open import foundation.sets +open import foundation.universe-levels +``` + +
+ +## Idea + +A **gaunt category** is a [category](category-theory.categories.md) such that +one of the following equivalent conditions hold: + +1. The + [isomorphism](category-theory.isomorphisms-in-categories.md)-[sets](foundation-core.sets.md) + are [propositions](foundation-core.propositions.md). +2. The objects form a set. +3. Every object is [rigid](category-theory.rigid-objects-categories.md), meaning + its [automorphism group](group-theory.automorphism-groups.md) is + [trivial](group-theory.trivial-groups.md). + +Gaunt categories forms the common intersection of (univalent) categories and +[strict categories](category-theory.strict-categories.md). We have the following +diagram relating the different notions of "category": + +```text + Gaunt categories + / \ + / \ + v v + Categories Strict categories + \ / + \ / + v v + Preunivalent categories + | + | + v + Precategories +``` + +## Definitions + +### The predicate on precategories that there is at most one isomorphism between any two objects + +```agda +module _ + {l1 l2 : Level} (C : Precategory l1 l2) + where + + is-prop-iso-prop-Precategory : Prop (l1 ⊔ l2) + is-prop-iso-prop-Precategory = + Π-Prop + ( obj-Precategory C) + ( λ x → + Π-Prop + ( obj-Precategory C) + ( λ y → is-prop-Prop (iso-Precategory C x y))) + + is-prop-iso-Precategory : UU (l1 ⊔ l2) + is-prop-iso-Precategory = type-Prop is-prop-iso-prop-Precategory + + is-property-is-prop-iso-Precategory : is-prop is-prop-iso-Precategory + is-property-is-prop-iso-Precategory = + is-prop-type-Prop is-prop-iso-prop-Precategory +``` + +### The predicate on precategories of being gaunt + +```agda +module _ + {l1 l2 : Level} (C : Precategory l1 l2) + where + + is-gaunt-prop-Precategory : Prop (l1 ⊔ l2) + is-gaunt-prop-Precategory = + prod-Prop (is-category-prop-Precategory C) (is-prop-iso-prop-Precategory C) + + is-gaunt-Precategory : UU (l1 ⊔ l2) + is-gaunt-Precategory = type-Prop is-gaunt-prop-Precategory +``` + +### The predicate on categories of being gaunt + +```agda +module _ + {l1 l2 : Level} (C : Category l1 l2) + where + + is-gaunt-prop-Category : Prop (l1 ⊔ l2) + is-gaunt-prop-Category = is-prop-iso-prop-Precategory (precategory-Category C) + + is-gaunt-Category : UU (l1 ⊔ l2) + is-gaunt-Category = type-Prop is-gaunt-prop-Category +``` + +### The type of gaunt categories + +```agda +Gaunt-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) +Gaunt-Category l1 l2 = Σ (Category l1 l2) (is-gaunt-Category) + +module _ + {l1 l2 : Level} (C : Gaunt-Category l1 l2) + where + + category-Gaunt-Category : Category l1 l2 + category-Gaunt-Category = pr1 C + + obj-Gaunt-Category : UU l1 + obj-Gaunt-Category = obj-Category category-Gaunt-Category + + hom-set-Gaunt-Category : + obj-Gaunt-Category → obj-Gaunt-Category → Set l2 + hom-set-Gaunt-Category = + hom-set-Category category-Gaunt-Category + + hom-Gaunt-Category : + obj-Gaunt-Category → obj-Gaunt-Category → UU l2 + hom-Gaunt-Category = hom-Category category-Gaunt-Category + + is-set-hom-Gaunt-Category : + (x y : obj-Gaunt-Category) → is-set (hom-Gaunt-Category x y) + is-set-hom-Gaunt-Category = + is-set-hom-Category category-Gaunt-Category + + comp-hom-Gaunt-Category : + {x y z : obj-Gaunt-Category} → + hom-Gaunt-Category y z → + hom-Gaunt-Category x y → + hom-Gaunt-Category x z + comp-hom-Gaunt-Category = + comp-hom-Category category-Gaunt-Category + + associative-comp-hom-Gaunt-Category : + {x y z w : obj-Gaunt-Category} + (h : hom-Gaunt-Category z w) + (g : hom-Gaunt-Category y z) + (f : hom-Gaunt-Category x y) → + comp-hom-Gaunt-Category (comp-hom-Gaunt-Category h g) f ＝ + comp-hom-Gaunt-Category h (comp-hom-Gaunt-Category g f) + associative-comp-hom-Gaunt-Category = + associative-comp-hom-Category category-Gaunt-Category + + associative-composition-operation-Gaunt-Category : + associative-composition-operation-binary-family-Set + hom-set-Gaunt-Category + associative-composition-operation-Gaunt-Category = + associative-composition-operation-Category + ( category-Gaunt-Category) + + id-hom-Gaunt-Category : + {x : obj-Gaunt-Category} → hom-Gaunt-Category x x + id-hom-Gaunt-Category = + id-hom-Category category-Gaunt-Category + + left-unit-law-comp-hom-Gaunt-Category : + {x y : obj-Gaunt-Category} (f : hom-Gaunt-Category x y) → + comp-hom-Gaunt-Category id-hom-Gaunt-Category f ＝ f + left-unit-law-comp-hom-Gaunt-Category = + left-unit-law-comp-hom-Category category-Gaunt-Category + + right-unit-law-comp-hom-Gaunt-Category : + {x y : obj-Gaunt-Category} (f : hom-Gaunt-Category x y) → + comp-hom-Gaunt-Category f id-hom-Gaunt-Category ＝ f + right-unit-law-comp-hom-Gaunt-Category = + right-unit-law-comp-hom-Category category-Gaunt-Category + + is-unital-composition-operation-Gaunt-Category : + is-unital-composition-operation-binary-family-Set + hom-set-Gaunt-Category + comp-hom-Gaunt-Category + is-unital-composition-operation-Gaunt-Category = + is-unital-composition-operation-Category + ( category-Gaunt-Category) + + is-gaunt-Gaunt-Category : + is-gaunt-Category category-Gaunt-Category + is-gaunt-Gaunt-Category = pr2 C +``` + +### The underlying nonunital precategory of a gaunt category + +```agda +nonunital-precategory-Gaunt-Category : + {l1 l2 : Level} → Gaunt-Category l1 l2 → Nonunital-Precategory l1 l2 +nonunital-precategory-Gaunt-Category C = + nonunital-precategory-Category (category-Gaunt-Category C) +``` + +### The underlying precategory of a gaunt category + +```agda +precategory-Gaunt-Category : + {l1 l2 : Level} → Gaunt-Category l1 l2 → Precategory l1 l2 +precategory-Gaunt-Category C = precategory-Category (category-Gaunt-Category C) +``` + +### The underlying preunivalent category of a gaunt category + +```agda +preunivalent-category-Gaunt-Category : + {l1 l2 : Level} → Gaunt-Category l1 l2 → Preunivalent-Category l1 l2 +preunivalent-category-Gaunt-Category C = + preunivalent-category-Category (category-Gaunt-Category C) +``` + +### The total hom-type of a gaunt category + +```agda +total-hom-Gaunt-Category : + {l1 l2 : Level} (C : Gaunt-Category l1 l2) → UU (l1 ⊔ l2) +total-hom-Gaunt-Category C = + total-hom-Category (category-Gaunt-Category C) + +obj-total-hom-Gaunt-Category : + {l1 l2 : Level} (C : Gaunt-Category l1 l2) → + total-hom-Gaunt-Category C → + obj-Gaunt-Category C × obj-Gaunt-Category C +obj-total-hom-Gaunt-Category C = + obj-total-hom-Category (category-Gaunt-Category C) +``` + +### Equalities induce morphisms + +```agda +module _ + {l1 l2 : Level} + (C : Gaunt-Category l1 l2) + where + + hom-eq-Gaunt-Category : + (x y : obj-Gaunt-Category C) → x ＝ y → hom-Gaunt-Category C x y + hom-eq-Gaunt-Category = + hom-eq-Category (category-Gaunt-Category C) + + hom-inv-eq-Gaunt-Category : + (x y : obj-Gaunt-Category C) → x ＝ y → hom-Gaunt-Category C y x + hom-inv-eq-Gaunt-Category = + hom-inv-eq-Category (category-Gaunt-Category C) +``` + +## Properties + +### Preunivalent categories whose isomorphism-sets are propositions are strict categories + +```agda +is-strict-category-is-prop-iso-Preunivalent-Category : + {l1 l2 : Level} (C : Preunivalent-Category l1 l2) → + is-prop-iso-Precategory (precategory-Preunivalent-Category C) → + is-strict-category-Preunivalent-Category C +is-strict-category-is-prop-iso-Preunivalent-Category C is-prop-iso-C x y = + is-prop-emb (emb-iso-eq-Preunivalent-Category C) (is-prop-iso-C x y) +``` + +### Gaunt categories are strict + +```agda +is-strict-category-is-gaunt-Category : + {l1 l2 : Level} (C : Category l1 l2) → + is-gaunt-Category C → is-strict-category-Category C +is-strict-category-is-gaunt-Category C = + is-strict-category-is-prop-iso-Preunivalent-Category + ( preunivalent-category-Category C) +``` + +### A strict category is gaunt if `iso-eq` is surjective + +```agda +module _ + {l1 l2 : Level} (C : Strict-Category l1 l2) + where + + is-category-is-surjective-iso-eq-Strict-Category : + is-surjective-iso-eq-Precategory (precategory-Strict-Category C) → + is-category-Precategory (precategory-Strict-Category C) + is-category-is-surjective-iso-eq-Strict-Category = + is-category-is-surjective-iso-eq-Preunivalent-Category + ( preunivalent-category-Strict-Category C) + + is-prop-iso-is-category-Strict-Category : + is-category-Precategory (precategory-Strict-Category C) → + is-prop-iso-Precategory (precategory-Strict-Category C) + is-prop-iso-is-category-Strict-Category is-category-C x y = + is-prop-is-equiv' (is-category-C x y) (is-set-obj-Strict-Category C x y) + + is-prop-iso-is-surjective-iso-eq-Strict-Category : + is-surjective-iso-eq-Precategory (precategory-Strict-Category C) → + is-prop-iso-Precategory (precategory-Strict-Category C) + is-prop-iso-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C = + is-prop-iso-is-category-Strict-Category + ( is-category-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C) + + is-gaunt-is-surjective-iso-eq-Strict-Category : + is-surjective-iso-eq-Precategory (precategory-Strict-Category C) → + is-gaunt-Precategory (precategory-Strict-Category C) + pr1 (is-gaunt-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C) = + is-category-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C + pr2 (is-gaunt-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C) = + is-prop-iso-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C +``` + +### A category is gaunt if and only if every object is rigid + +**Proof:** Using the fact that a type is a +[proposition](foundation-core.propositions.md) if and only if having an +inhabitant implies it is [contractible](foundation-core.contractible-types.md), +we can apply +[isomorphism induction](category-theory.isomorphism-induction-categories.md) to +get our result. + +```agda +module _ + {l1 l2 : Level} (C : Category l1 l2) + where + + is-gaunt-is-rigid-Category : + ((x : obj-Category C) → is-rigid-obj-Category C x) → is-gaunt-Category C + is-gaunt-is-rigid-Category is-rigid-obj-C x y = + is-prop-is-proof-irrelevant (ind-iso-Category C _ (is-rigid-obj-C x)) + + is-rigid-is-gaunt-Category : + is-gaunt-Category C → (x : obj-Category C) → is-rigid-obj-Category C x + is-rigid-is-gaunt-Category is-gaunt-C x = + is-proof-irrelevant-is-prop (is-gaunt-C x x) (id-iso-Category C) +``` + +## See also + +- [Posets](order-theory.posets.md) are gaunt. + +## External links + +- [Gaunt (pre)categories](https://1lab.dev/Cat.Gaunt.html) at 1lab +- [gaunt category](https://ncatlab.org/nlab/show/gaunt+category#in_type_theory) + at nlab diff --git a/src/category-theory/isomorphism-induction-categories.lagda.md b/src/category-theory/isomorphism-induction-categories.lagda.md index 5a9dd08498..a736c05d8a 100644 --- a/src/category-theory/isomorphism-induction-categories.lagda.md +++ b/src/category-theory/isomorphism-induction-categories.lagda.md @@ -29,8 +29,8 @@ open import foundation.universe-levels ## Idea -**Isomorphism induction** in a category `𝒞` is the principle asserting that, -given an object `A : 𝒞` and any type family +**Isomorphism induction** in a [category](category-theory.categories.md) `𝒞` is +the principle asserting that, given an object `A : 𝒞` and any type family ```text P : (B : 𝒞) (ϕ : A ≅ B) → 𝒰 diff --git a/src/category-theory/isomorphisms-in-categories.lagda.md b/src/category-theory/isomorphisms-in-categories.lagda.md index 4937a6170b..ed3292cb95 100644 --- a/src/category-theory/isomorphisms-in-categories.lagda.md +++ b/src/category-theory/isomorphisms-in-categories.lagda.md @@ -557,13 +557,15 @@ module _ {x y : obj-Category C} where - is-prop-iso-Category : + is-prop-iso-is-prop-hom-Category : is-prop (hom-Category C x y) → is-prop (iso-Category C x y) - is-prop-iso-Category = is-prop-iso-Precategory (precategory-Category C) + is-prop-iso-is-prop-hom-Category = + is-prop-iso-is-prop-hom-Precategory (precategory-Category C) - iso-prop-Category : + iso-prop-is-prop-hom-Category : is-prop (hom-Category C x y) → Prop l2 - iso-prop-Category = iso-prop-Precategory (precategory-Category C) + iso-prop-is-prop-hom-Category = + iso-prop-is-prop-hom-Precategory (precategory-Category C) ``` ### When `hom x y` and `hom y x` are propositions, it suffices to provide a morphism in each direction to construct an isomorphism diff --git a/src/category-theory/isomorphisms-in-precategories.lagda.md b/src/category-theory/isomorphisms-in-precategories.lagda.md index dce1b6d1aa..3b06e59939 100644 --- a/src/category-theory/isomorphisms-in-precategories.lagda.md +++ b/src/category-theory/isomorphisms-in-precategories.lagda.md @@ -14,6 +14,7 @@ open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-types +open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps open import foundation.propositions @@ -432,6 +433,12 @@ module _ pr1 (inv-iso-Precategory f) = hom-inv-iso-Precategory C f pr2 (inv-iso-Precategory f) = is-iso-inv-is-iso-Precategory C (is-iso-iso-Precategory C f) + + is-iso-inv-iso-Precategory : + (f : iso-Precategory C x y) → + is-iso-Precategory C (hom-inv-iso-Precategory C f) + is-iso-inv-iso-Precategory f = + is-iso-iso-Precategory C (inv-iso-Precategory f) ``` ### Groupoid laws of isomorphisms in precategories @@ -519,6 +526,33 @@ module _ ( is-section-hom-inv-iso-Precategory C f) ``` +### The inverse operation is a fibered involution on isomorphisms + +```agda +module _ + {l1 l2 : Level} + (C : Precategory l1 l2) + where + + is-fibered-involution-inv-iso-Precategory : + {x y : obj-Precategory C} → + inv-iso-Precategory C {y} {x} ∘ inv-iso-Precategory C {x} {y} ~ id + is-fibered-involution-inv-iso-Precategory f = refl + + is-equiv-inv-iso-Precategory : + {x y : obj-Precategory C} → is-equiv (inv-iso-Precategory C {x} {y}) + is-equiv-inv-iso-Precategory = + is-equiv-is-invertible + ( inv-iso-Precategory C) + ( is-fibered-involution-inv-iso-Precategory) + ( is-fibered-involution-inv-iso-Precategory) + + equiv-inv-iso-Precategory : + {x y : obj-Precategory C} → iso-Precategory C x y ≃ iso-Precategory C y x + pr1 equiv-inv-iso-Precategory = inv-iso-Precategory C + pr2 equiv-inv-iso-Precategory = is-equiv-inv-iso-Precategory +``` + ### A morphism `f` is an isomorphism if and only if precomposition by `f` is an equivalence **Proof:** If `f` is an isomorphism with inverse `f⁻¹`, then precomposing with @@ -686,15 +720,17 @@ module _ {x y : obj-Precategory C} where - is-prop-iso-Precategory : + is-prop-iso-is-prop-hom-Precategory : is-prop (hom-Precategory C x y) → is-prop (iso-Precategory C x y) - is-prop-iso-Precategory = is-prop-type-subtype (is-iso-prop-Precategory C) + is-prop-iso-is-prop-hom-Precategory = + is-prop-type-subtype (is-iso-prop-Precategory C) - iso-prop-Precategory : + iso-prop-is-prop-hom-Precategory : is-prop (hom-Precategory C x y) → Prop l2 - pr1 (iso-prop-Precategory _) = iso-Precategory C x y - pr2 (iso-prop-Precategory is-prop-hom-C-x-y) = - is-prop-iso-Precategory is-prop-hom-C-x-y + pr1 (iso-prop-is-prop-hom-Precategory is-prop-hom-C-x-y) = + iso-Precategory C x y + pr2 (iso-prop-is-prop-hom-Precategory is-prop-hom-C-x-y) = + is-prop-iso-is-prop-hom-Precategory is-prop-hom-C-x-y ``` ### When `hom x y` and `hom y x` are propositions, it suffices to provide a morphism in each direction to construct an isomorphism diff --git a/src/category-theory/isomorphisms-in-subprecategories.lagda.md b/src/category-theory/isomorphisms-in-subprecategories.lagda.md new file mode 100644 index 0000000000..e0d8560de0 --- /dev/null +++ b/src/category-theory/isomorphisms-in-subprecategories.lagda.md @@ -0,0 +1,264 @@ +# Isomorphisms in subprecategories + +```agda +module category-theory.isomorphisms-in-subprecategories where +``` + +
Imports + +```agda +open import category-theory.isomorphisms-in-precategories +open import category-theory.precategories +open import category-theory.subprecategories + +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.propositions +open import foundation.sets +open import foundation.subtypes +open import foundation.universe-levels +``` + +
+ +## Definitions + +### Isomorphisms in subprecategories + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + {x y : obj-Subprecategory C P} + (f : hom-Subprecategory C P x y) + where + + is-iso-prop-Subprecategory : Prop (l2 ⊔ l4) + is-iso-prop-Subprecategory = + is-iso-prop-Precategory (precategory-Subprecategory C P) {x} {y} f + + is-iso-Subprecategory : UU (l2 ⊔ l4) + is-iso-Subprecategory = type-Prop is-iso-prop-Subprecategory + + is-prop-is-iso-Subprecategory : is-prop is-iso-Subprecategory + is-prop-is-iso-Subprecategory = is-prop-type-Prop is-iso-prop-Subprecategory +``` + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + (x y : obj-Subprecategory C P) + where + + iso-set-Subprecategory : Set (l2 ⊔ l4) + iso-set-Subprecategory = + iso-set-Precategory (precategory-Subprecategory C P) {x} {y} + + iso-Subprecategory : UU (l2 ⊔ l4) + iso-Subprecategory = type-Set iso-set-Subprecategory + + is-set-iso-Subprecategory : is-set iso-Subprecategory + is-set-iso-Subprecategory = is-set-type-Set iso-set-Subprecategory +``` + +#### The predicate on an isomorphism proof of being contained in a subprecategory + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + {x y : obj-Subprecategory C P} + (f : hom-Subprecategory C P x y) + (is-iso-f : is-iso-Precategory C (inclusion-hom-Subprecategory C P x y f)) + where + + is-in-is-iso-prop-Subprecategory : Prop l4 + is-in-is-iso-prop-Subprecategory = + subtype-hom-obj-subprecategory-Subprecategory C P y x + ( hom-inv-is-iso-Precategory C is-iso-f) + + is-in-is-iso-Subprecategory : UU l4 + is-in-is-iso-Subprecategory = + type-Prop is-in-is-iso-prop-Subprecategory + + is-prop-is-in-is-iso-Subprecategory : is-prop is-in-is-iso-Subprecategory + is-prop-is-in-is-iso-Subprecategory = + is-prop-type-Prop is-in-is-iso-prop-Subprecategory + + is-iso-is-in-is-iso-Subprecategory : + is-in-is-iso-Subprecategory → is-iso-Subprecategory C P f + pr1 (pr1 (is-iso-is-in-is-iso-Subprecategory is-in-is-iso-f)) = + hom-inv-is-iso-Precategory C is-iso-f + pr2 (pr1 (is-iso-is-in-is-iso-Subprecategory is-in-is-iso-f)) = + is-in-is-iso-f + pr1 (pr2 (is-iso-is-in-is-iso-Subprecategory is-in-is-iso-f)) = + eq-type-subtype + ( subtype-hom-obj-subprecategory-Subprecategory C P y y) + ( pr1 (pr2 is-iso-f)) + pr2 (pr2 (is-iso-is-in-is-iso-Subprecategory is-in-is-iso-f)) = + eq-type-subtype + ( subtype-hom-obj-subprecategory-Subprecategory C P x x) + ( pr2 (pr2 is-iso-f)) +``` + +#### The predicate on an isomorphism between objects in the subprecategory of being contained in the subprecategory + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + {x y : obj-Subprecategory C P} + (f : + iso-Precategory C + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y)) + where + + is-in-iso-obj-subprecategory-prop-Subprecategory : Prop l4 + is-in-iso-obj-subprecategory-prop-Subprecategory = + Σ-Prop + ( subtype-hom-obj-subprecategory-Subprecategory C P x y + ( hom-iso-Precategory C f)) + ( λ f₀ → + is-in-is-iso-prop-Subprecategory C P + ( hom-iso-Precategory C f , f₀) + ( is-iso-iso-Precategory C f)) + + is-in-iso-obj-subprecategory-Subprecategory : UU l4 + is-in-iso-obj-subprecategory-Subprecategory = + type-Prop is-in-iso-obj-subprecategory-prop-Subprecategory + + is-prop-is-in-iso-obj-subprecategory-Subprecategory : + is-prop is-in-iso-obj-subprecategory-Subprecategory + is-prop-is-in-iso-obj-subprecategory-Subprecategory = + is-prop-type-Prop is-in-iso-obj-subprecategory-prop-Subprecategory + + is-iso-is-in-iso-obj-subprecategory-Subprecategory : + ((f₀ , f₁) : is-in-iso-obj-subprecategory-Subprecategory) → + is-iso-Subprecategory C P (hom-iso-Precategory C f , f₀) + is-iso-is-in-iso-obj-subprecategory-Subprecategory (f₀ , f₁) = + is-iso-is-in-is-iso-Subprecategory C P (pr1 f , f₀) (pr2 f) f₁ + + iso-is-in-iso-obj-subprecategory-Subprecategory : + is-in-iso-obj-subprecategory-Subprecategory → iso-Subprecategory C P x y + pr1 (pr1 (iso-is-in-iso-obj-subprecategory-Subprecategory is-in-iso-f)) = + hom-iso-Precategory C f + pr2 (pr1 (iso-is-in-iso-obj-subprecategory-Subprecategory is-in-iso-f)) = + pr1 is-in-iso-f + pr2 (iso-is-in-iso-obj-subprecategory-Subprecategory is-in-iso-f) = + is-iso-is-in-is-iso-Subprecategory C P _ + ( is-iso-iso-Precategory C f) + ( pr2 is-in-iso-f) +``` + +#### The predicate on an isomorphism between any objects of being contained in the subprecategory + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + {x y : obj-Precategory C} + (f : iso-Precategory C x y) + where + + is-in-iso-prop-Subprecategory : Prop (l3 ⊔ l4) + is-in-iso-prop-Subprecategory = + Σ-Prop + ( subtype-obj-Subprecategory C P x) + ( λ x₀ → + Σ-Prop + ( subtype-obj-Subprecategory C P y) + ( λ y₀ → + is-in-iso-obj-subprecategory-prop-Subprecategory C P + { x , x₀} {y , y₀} f)) + + is-in-iso-Subprecategory : UU (l3 ⊔ l4) + is-in-iso-Subprecategory = type-Prop is-in-iso-prop-Subprecategory + + is-prop-is-in-iso-Subprecategory : is-prop is-in-iso-Subprecategory + is-prop-is-in-iso-Subprecategory = + is-prop-type-Prop is-in-iso-prop-Subprecategory + + iso-is-in-iso-Subprecategory : + (is-in-iso-f : is-in-iso-Subprecategory) → + iso-Subprecategory C P (x , pr1 is-in-iso-f) (y , pr1 (pr2 is-in-iso-f)) + iso-is-in-iso-Subprecategory is-in-iso-f = + iso-is-in-iso-obj-subprecategory-Subprecategory C P f + ( pr2 (pr2 is-in-iso-f)) + + is-iso-is-in-iso-Subprecategory : + ( is-in-iso-f : is-in-iso-Subprecategory) → + is-iso-Subprecategory C P + ( pr1 f , pr2 (pr1 (iso-is-in-iso-Subprecategory is-in-iso-f))) + is-iso-is-in-iso-Subprecategory is-in-iso-f = + pr2 (iso-is-in-iso-Subprecategory is-in-iso-f) +``` + +### If a subprecategory contains an object, it contains its identity ismorphism + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + (x : obj-Subprecategory C P) + where + + is-iso-id-hom-Subprecategory : + is-iso-Subprecategory C P (id-hom-Subprecategory C P {x}) + is-iso-id-hom-Subprecategory = + is-iso-id-hom-Precategory (precategory-Subprecategory C P) + + is-in-is-iso-id-obj-subprecategory-Subprecategory : + is-in-is-iso-Subprecategory C P {x} + (id-hom-Subprecategory C P) (is-iso-id-hom-Precategory C) + is-in-is-iso-id-obj-subprecategory-Subprecategory = + contains-id-Subprecategory C P + ( inclusion-obj-Subprecategory C P x) + ( is-in-obj-inclusion-obj-Subprecategory C P x) + + is-in-iso-id-obj-subprecategory-Subprecategory : + is-in-iso-obj-subprecategory-Subprecategory C P (id-iso-Precategory C) + pr1 is-in-iso-id-obj-subprecategory-Subprecategory = + is-in-is-iso-id-obj-subprecategory-Subprecategory + pr2 is-in-iso-id-obj-subprecategory-Subprecategory = + is-in-is-iso-id-obj-subprecategory-Subprecategory + + is-in-iso-id-Subprecategory : + is-in-iso-Subprecategory C P (id-iso-Precategory C) + pr1 is-in-iso-id-Subprecategory = is-in-obj-inclusion-obj-Subprecategory C P x + pr1 (pr2 is-in-iso-id-Subprecategory) = + is-in-obj-inclusion-obj-Subprecategory C P x + pr2 (pr2 is-in-iso-id-Subprecategory) = + is-in-iso-id-obj-subprecategory-Subprecategory +``` + +## Properties + +### Isomorphisms in a subprecategory are isomorphisms in the base category + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + {x y : obj-Subprecategory C P} + where + + is-iso-base-is-iso-Subprecategory : + {f : hom-Subprecategory C P x y} → + is-iso-Subprecategory C P f → + is-iso-Precategory C (inclusion-hom-Subprecategory C P x y f) + pr1 (is-iso-base-is-iso-Subprecategory is-iso-f) = + pr1 (pr1 is-iso-f) + pr1 (pr2 (is-iso-base-is-iso-Subprecategory is-iso-f)) = + ap pr1 (pr1 (pr2 (is-iso-f))) + pr2 (pr2 (is-iso-base-is-iso-Subprecategory is-iso-f)) = + ap pr1 (pr2 (pr2 (is-iso-f))) +``` diff --git a/src/category-theory/large-subprecategories.lagda.md b/src/category-theory/large-subprecategories.lagda.md index ecba71cf2f..efeeebbc58 100644 --- a/src/category-theory/large-subprecategories.lagda.md +++ b/src/category-theory/large-subprecategories.lagda.md @@ -89,8 +89,8 @@ module _ (P : Large-Subprecategory γ δ C) where - is-in-subtype-obj-Large-Subprecategory : + is-in-obj-Large-Subprecategory : {l : Level} → obj-Large-Precategory C l → UU (γ l) - is-in-subtype-obj-Large-Subprecategory = + is-in-obj-Large-Subprecategory = is-in-subtype (subtype-obj-Large-Subprecategory P _) ``` diff --git a/src/category-theory/natural-isomorphisms-functors-categories.lagda.md b/src/category-theory/natural-isomorphisms-functors-categories.lagda.md index 74a02c09b1..f8f8894aea 100644 --- a/src/category-theory/natural-isomorphisms-functors-categories.lagda.md +++ b/src/category-theory/natural-isomorphisms-functors-categories.lagda.md @@ -195,7 +195,7 @@ module _ ( hom-family-natural-transformation-Category C D F G ( natural-transformation-natural-isomorphism-Category)) coherence-square-natural-isomorphism-Category = - coherence-square-natural-transformation-Category C D F G + naturality-natural-transformation-Category C D F G ( natural-transformation-natural-isomorphism-Category) is-natural-isomorphism-natural-isomorphism-Category : diff --git a/src/category-theory/natural-isomorphisms-functors-precategories.lagda.md b/src/category-theory/natural-isomorphisms-functors-precategories.lagda.md index a7ca42e597..35a1b71552 100644 --- a/src/category-theory/natural-isomorphisms-functors-precategories.lagda.md +++ b/src/category-theory/natural-isomorphisms-functors-precategories.lagda.md @@ -177,7 +177,7 @@ module _ ( hom-family-natural-transformation-Precategory C D F G ( natural-transformation-natural-isomorphism-Precategory)) coherence-square-natural-isomorphism-Precategory = - coherence-square-natural-transformation-Precategory C D F G + naturality-natural-transformation-Precategory C D F G ( natural-transformation-natural-isomorphism-Precategory) is-natural-isomorphism-natural-isomorphism-Precategory : diff --git a/src/category-theory/natural-isomorphisms-maps-categories.lagda.md b/src/category-theory/natural-isomorphisms-maps-categories.lagda.md index af45614ba5..571c2723dc 100644 --- a/src/category-theory/natural-isomorphisms-maps-categories.lagda.md +++ b/src/category-theory/natural-isomorphisms-maps-categories.lagda.md @@ -194,7 +194,7 @@ module _ ( hom-family-natural-transformation-map-Category C D F G ( natural-transformation-map-natural-isomorphism-map-Category)) coherence-square-natural-isomorphism-map-Category = - coherence-square-natural-transformation-map-Category C D F G + naturality-natural-transformation-map-Category C D F G ( natural-transformation-map-natural-isomorphism-map-Category) is-natural-isomorphism-map-natural-isomorphism-map-Category : diff --git a/src/category-theory/natural-isomorphisms-maps-precategories.lagda.md b/src/category-theory/natural-isomorphisms-maps-precategories.lagda.md index 420e96b26f..1e2ff1ccc6 100644 --- a/src/category-theory/natural-isomorphisms-maps-precategories.lagda.md +++ b/src/category-theory/natural-isomorphisms-maps-precategories.lagda.md @@ -161,7 +161,7 @@ module _ ( hom-family-natural-transformation-map-Precategory C D F G ( natural-transformation-map-natural-isomorphism-map-Precategory)) coherence-square-natural-isomorphism-map-Precategory = - coherence-square-natural-transformation-map-Precategory C D F G + naturality-natural-transformation-map-Precategory C D F G ( natural-transformation-map-natural-isomorphism-map-Precategory) is-natural-isomorphism-map-natural-isomorphism-map-Precategory : @@ -415,7 +415,7 @@ module _ ( ap ( comp-hom-Precategory' D _) ( inv - ( coherence-square-natural-transformation-map-Precategory + ( naturality-natural-transformation-map-Precategory C D F G f g))) ∙ ( associative-comp-hom-Precategory D ( hom-map-Precategory C D G g) diff --git a/src/category-theory/natural-transformations-functors-categories.lagda.md b/src/category-theory/natural-transformations-functors-categories.lagda.md index 66303411bd..31ddf1de0b 100644 --- a/src/category-theory/natural-transformations-functors-categories.lagda.md +++ b/src/category-theory/natural-transformations-functors-categories.lagda.md @@ -74,12 +74,12 @@ module _ ( F) ( G) - coherence-square-natural-transformation-Category : + naturality-natural-transformation-Category : (γ : natural-transformation-Category) → is-natural-transformation-Category ( hom-family-natural-transformation-Category γ) - coherence-square-natural-transformation-Category = - coherence-square-natural-transformation-Precategory + naturality-natural-transformation-Category = + naturality-natural-transformation-Precategory ( precategory-Category C) ( precategory-Category D) ( F) diff --git a/src/category-theory/natural-transformations-functors-from-small-to-large-categories.lagda.md b/src/category-theory/natural-transformations-functors-from-small-to-large-categories.lagda.md new file mode 100644 index 0000000000..53da547b30 --- /dev/null +++ b/src/category-theory/natural-transformations-functors-from-small-to-large-categories.lagda.md @@ -0,0 +1,245 @@ +# Natural transformations between functors from small to large categories + +```agda +module category-theory.natural-transformations-functors-from-small-to-large-categories where +``` + +
Imports + +```agda +open import category-theory.categories +open import category-theory.functors-from-small-to-large-categories +open import category-theory.large-categories +open import category-theory.natural-transformations-functors-from-small-to-large-precategories + +open import foundation.dependent-pair-types +open import foundation.embeddings +open import foundation.equivalences +open import foundation.homotopies +open import foundation.identity-types +open import foundation.propositions +open import foundation.sets +open import foundation.universe-levels +``` + +
+ +## Idea + +Given a small [category](category-theory.categories.md) `C` and a +[large category](category-theory.large-categories.md) `D`, a **natural +transformation** from a +[functor](category-theory.functors-from-small-to-large-categories.md) +`F : C → D` to `G : C → D` consists of : + +- a family of morphisms `a : (x : C) → hom (F x) (G x)` such that the following + identity holds: +- `(G f) ∘ (a x) = (a y) ∘ (F f)`, for all `f : hom x y`. + +## Definition + +```agda +module _ + {l1 l2 γF γG : Level} {α : Level → Level} {β : Level → Level → Level} + (C : Category l1 l2) + (D : Large-Category α β) + (F : functor-Small-Large-Category C D γF) + (G : functor-Small-Large-Category C D γG) + where + + hom-family-functor-Small-Large-Category : UU (l1 ⊔ β γF γG) + hom-family-functor-Small-Large-Category = + hom-family-functor-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + is-natural-transformation-Small-Large-Category : + hom-family-functor-Small-Large-Category → UU (l1 ⊔ l2 ⊔ β γF γG) + is-natural-transformation-Small-Large-Category = + is-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + natural-transformation-Small-Large-Category : UU (l1 ⊔ l2 ⊔ β γF γG) + natural-transformation-Small-Large-Category = + natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + hom-family-natural-transformation-Small-Large-Category : + natural-transformation-Small-Large-Category → + hom-family-functor-Small-Large-Category + hom-family-natural-transformation-Small-Large-Category = + hom-family-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + naturality-natural-transformation-Small-Large-Category : + (γ : natural-transformation-Small-Large-Category) → + is-natural-transformation-Small-Large-Category + ( hom-family-natural-transformation-Small-Large-Category γ) + naturality-natural-transformation-Small-Large-Category = + naturality-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G +``` + +## Composition and identity of natural transformations + +```agda +module _ + {l1 l2 : Level} {α : Level → Level} {β : Level → Level → Level} + (C : Category l1 l2) + (D : Large-Category α β) + where + + id-natural-transformation-Small-Large-Category : + {γF : Level} (F : functor-Small-Large-Category C D γF) → + natural-transformation-Small-Large-Category C D F F + id-natural-transformation-Small-Large-Category = + id-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) + + comp-natural-transformation-Small-Large-Category : + {γF γG γH : Level} + (F : functor-Small-Large-Category C D γF) + (G : functor-Small-Large-Category C D γG) + (H : functor-Small-Large-Category C D γH) → + natural-transformation-Small-Large-Category C D G H → + natural-transformation-Small-Large-Category C D F G → + natural-transformation-Small-Large-Category C D F H + comp-natural-transformation-Small-Large-Category = + comp-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) +``` + +## Properties + +### That a family of morphisms is a natural transformation is a proposition + +This follows from the fact that the hom-types are +[sets](foundation-core.sets.md). + +```agda +module _ + {l1 l2 γF γG : Level} {α : Level → Level} {β : Level → Level → Level} + (C : Category l1 l2) + (D : Large-Category α β) + (F : functor-Small-Large-Category C D γF) + (G : functor-Small-Large-Category C D γG) + where + + is-prop-is-natural-transformation-Small-Large-Category : + (γ : hom-family-functor-Small-Large-Category C D F G) → + is-prop (is-natural-transformation-Small-Large-Category C D F G γ) + is-prop-is-natural-transformation-Small-Large-Category = + is-prop-is-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + is-natural-transformation-prop-Small-Large-Category : + (γ : hom-family-functor-Small-Large-Category C D F G) → + Prop (l1 ⊔ l2 ⊔ β γF γG) + is-natural-transformation-prop-Small-Large-Category = + is-natural-transformation-prop-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G +``` + +### The set of natural transformations + +```agda +module _ + {l1 l2 γF γG : Level} {α : Level → Level} {β : Level → Level → Level} + (C : Category l1 l2) + (D : Large-Category α β) + (F : functor-Small-Large-Category C D γF) + (G : functor-Small-Large-Category C D γG) + where + + is-emb-hom-family-natural-transformation-Small-Large-Category : + is-emb (hom-family-natural-transformation-Small-Large-Category C D F G) + is-emb-hom-family-natural-transformation-Small-Large-Category = + is-emb-hom-family-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + emb-hom-family-natural-transformation-Small-Large-Category : + natural-transformation-Small-Large-Category C D F G ↪ + hom-family-functor-Small-Large-Category C D F G + emb-hom-family-natural-transformation-Small-Large-Category = + emb-hom-family-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + is-set-natural-transformation-Small-Large-Category : + is-set (natural-transformation-Small-Large-Category C D F G) + is-set-natural-transformation-Small-Large-Category = + is-set-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + natural-transformation-set-Small-Large-Category : + Set (l1 ⊔ l2 ⊔ β γF γG) + pr1 (natural-transformation-set-Small-Large-Category) = + natural-transformation-Small-Large-Category C D F G + pr2 (natural-transformation-set-Small-Large-Category) = + is-set-natural-transformation-Small-Large-Category + + extensionality-natural-transformation-Small-Large-Category : + (a b : natural-transformation-Small-Large-Category C D F G) → + ( a ＝ b) ≃ + ( hom-family-natural-transformation-Small-Large-Category C D F G a ~ + hom-family-natural-transformation-Small-Large-Category C D F G b) + extensionality-natural-transformation-Small-Large-Category = + extensionality-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G + + eq-htpy-hom-family-natural-transformation-Small-Large-Category : + (a b : natural-transformation-Small-Large-Category C D F G) → + ( hom-family-natural-transformation-Small-Large-Category C D F G a ~ + hom-family-natural-transformation-Small-Large-Category C D F G b) → + a ＝ b + eq-htpy-hom-family-natural-transformation-Small-Large-Category = + eq-htpy-hom-family-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) F G +``` + +### Categorical laws for natural transformations + +```agda +module _ + {l1 l2 : Level} {α : Level → Level} {β : Level → Level → Level} + (C : Category l1 l2) + (D : Large-Category α β) + where + + right-unit-law-comp-natural-transformation-Small-Large-Category : + {γF γG : Level} + (F : functor-Small-Large-Category C D γF) + (G : functor-Small-Large-Category C D γG) + (a : natural-transformation-Small-Large-Category C D F G) → + comp-natural-transformation-Small-Large-Category C D F F G a + ( id-natural-transformation-Small-Large-Category C D F) ＝ a + right-unit-law-comp-natural-transformation-Small-Large-Category = + right-unit-law-comp-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) + + left-unit-law-comp-natural-transformation-Small-Large-Category : + {γF γG : Level} + (F : functor-Small-Large-Category C D γF) + (G : functor-Small-Large-Category C D γG) + (a : natural-transformation-Small-Large-Category C D F G) → + comp-natural-transformation-Small-Large-Category C D F G G + ( id-natural-transformation-Small-Large-Category C D G) a ＝ a + left-unit-law-comp-natural-transformation-Small-Large-Category = + left-unit-law-comp-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) + + associative-comp-natural-transformation-Small-Large-Category : + {γF γG γH γI : Level} + (F : functor-Small-Large-Category C D γF) + (G : functor-Small-Large-Category C D γG) + (H : functor-Small-Large-Category C D γH) + (I : functor-Small-Large-Category C D γI) + (a : natural-transformation-Small-Large-Category C D F G) + (b : natural-transformation-Small-Large-Category C D G H) + (c : natural-transformation-Small-Large-Category C D H I) → + comp-natural-transformation-Small-Large-Category C D F G I + ( comp-natural-transformation-Small-Large-Category C D G H I c b) a ＝ + comp-natural-transformation-Small-Large-Category C D F H I c + ( comp-natural-transformation-Small-Large-Category C D F G H b a) + associative-comp-natural-transformation-Small-Large-Category = + associative-comp-natural-transformation-Small-Large-Precategory + ( precategory-Category C) (large-precategory-Large-Category D) +``` diff --git a/src/category-theory/natural-transformations-functors-from-small-to-large-precategories.lagda.md b/src/category-theory/natural-transformations-functors-from-small-to-large-precategories.lagda.md index 1df426695b..b74325623b 100644 --- a/src/category-theory/natural-transformations-functors-from-small-to-large-precategories.lagda.md +++ b/src/category-theory/natural-transformations-functors-from-small-to-large-precategories.lagda.md @@ -74,12 +74,12 @@ module _ ( map-functor-Small-Large-Precategory C D F) ( map-functor-Small-Large-Precategory C D G) - coherence-square-natural-transformation-Small-Large-Precategory : + naturality-natural-transformation-Small-Large-Precategory : (γ : natural-transformation-Small-Large-Precategory) → is-natural-transformation-Small-Large-Precategory ( hom-family-natural-transformation-Small-Large-Precategory γ) - coherence-square-natural-transformation-Small-Large-Precategory = - coherence-square-natural-transformation-map-Small-Large-Precategory C D + naturality-natural-transformation-Small-Large-Precategory = + naturality-natural-transformation-map-Small-Large-Precategory C D ( map-functor-Small-Large-Precategory C D F) ( map-functor-Small-Large-Precategory C D G) ``` diff --git a/src/category-theory/natural-transformations-functors-precategories.lagda.md b/src/category-theory/natural-transformations-functors-precategories.lagda.md index fc103c786c..b2e8a178af 100644 --- a/src/category-theory/natural-transformations-functors-precategories.lagda.md +++ b/src/category-theory/natural-transformations-functors-precategories.lagda.md @@ -69,12 +69,12 @@ module _ ( map-functor-Precategory C D F) ( map-functor-Precategory C D G) - coherence-square-natural-transformation-Precategory : + naturality-natural-transformation-Precategory : (γ : natural-transformation-Precategory) → is-natural-transformation-Precategory ( hom-family-natural-transformation-Precategory γ) - coherence-square-natural-transformation-Precategory = - coherence-square-natural-transformation-map-Precategory C D + naturality-natural-transformation-Precategory = + naturality-natural-transformation-map-Precategory C D ( map-functor-Precategory C D F) ( map-functor-Precategory C D G) ``` diff --git a/src/category-theory/natural-transformations-maps-categories.lagda.md b/src/category-theory/natural-transformations-maps-categories.lagda.md index 8cb719604a..04e78aa05e 100644 --- a/src/category-theory/natural-transformations-maps-categories.lagda.md +++ b/src/category-theory/natural-transformations-maps-categories.lagda.md @@ -74,12 +74,12 @@ module _ ( F) ( G) - coherence-square-natural-transformation-map-Category : + naturality-natural-transformation-map-Category : (γ : natural-transformation-map-Category) → is-natural-transformation-map-Category ( hom-family-natural-transformation-map-Category γ) - coherence-square-natural-transformation-map-Category = - coherence-square-natural-transformation-map-Precategory + naturality-natural-transformation-map-Category = + naturality-natural-transformation-map-Precategory ( precategory-Category C) ( precategory-Category D) ( F) diff --git a/src/category-theory/natural-transformations-maps-from-small-to-large-precategories.lagda.md b/src/category-theory/natural-transformations-maps-from-small-to-large-precategories.lagda.md index 1e5424b966..054aa20477 100644 --- a/src/category-theory/natural-transformations-maps-from-small-to-large-precategories.lagda.md +++ b/src/category-theory/natural-transformations-maps-from-small-to-large-precategories.lagda.md @@ -58,10 +58,10 @@ module _ ( obj-map-Small-Large-Precategory C D F x) ( obj-map-Small-Large-Precategory C D G x) - coherence-square-hom-family-map-Small-Large-Precategory : + naturality-hom-family-map-Small-Large-Precategory : hom-family-map-Small-Large-Precategory → {x y : obj-Precategory C} (f : hom-Precategory C x y) → UU (β γF γG) - coherence-square-hom-family-map-Small-Large-Precategory γ {x} {y} f = + naturality-hom-family-map-Small-Large-Precategory γ {x} {y} f = coherence-square-hom-Large-Precategory D ( hom-map-Small-Large-Precategory C D F f) ( γ x) @@ -72,7 +72,7 @@ module _ hom-family-map-Small-Large-Precategory → UU (l1 ⊔ l2 ⊔ β γF γG) is-natural-transformation-map-Small-Large-Precategory γ = {x y : obj-Precategory C} (f : hom-Precategory C x y) → - coherence-square-hom-family-map-Small-Large-Precategory γ f + naturality-hom-family-map-Small-Large-Precategory γ f natural-transformation-map-Small-Large-Precategory : UU (l1 ⊔ l2 ⊔ β γF γG) natural-transformation-map-Small-Large-Precategory = @@ -84,11 +84,11 @@ module _ hom-family-map-Small-Large-Precategory hom-family-natural-transformation-map-Small-Large-Precategory = pr1 - coherence-square-natural-transformation-map-Small-Large-Precategory : + naturality-natural-transformation-map-Small-Large-Precategory : (γ : natural-transformation-map-Small-Large-Precategory) → is-natural-transformation-map-Small-Large-Precategory ( hom-family-natural-transformation-map-Small-Large-Precategory γ) - coherence-square-natural-transformation-map-Small-Large-Precategory = pr2 + naturality-natural-transformation-map-Small-Large-Precategory = pr2 ``` ## Composition and identity of natural transformations @@ -140,7 +140,7 @@ module _ ( comp-hom-Large-Precategory' D ( hom-family-natural-transformation-map-Small-Large-Precategory C D F G α X)) - ( coherence-square-natural-transformation-map-Small-Large-Precategory + ( naturality-natural-transformation-map-Small-Large-Precategory C D G H β f)) ∙ ( associative-comp-hom-Large-Precategory D ( hom-family-natural-transformation-map-Small-Large-Precategory @@ -152,7 +152,7 @@ module _ ( comp-hom-Large-Precategory D ( hom-family-natural-transformation-map-Small-Large-Precategory C D G H β Y)) - ( coherence-square-natural-transformation-map-Small-Large-Precategory + ( naturality-natural-transformation-map-Small-Large-Precategory C D F G α f)) ∙ ( inv ( associative-comp-hom-Large-Precategory D diff --git a/src/category-theory/natural-transformations-maps-precategories.lagda.md b/src/category-theory/natural-transformations-maps-precategories.lagda.md index 060154f9a0..f130e7ef49 100644 --- a/src/category-theory/natural-transformations-maps-precategories.lagda.md +++ b/src/category-theory/natural-transformations-maps-precategories.lagda.md @@ -55,10 +55,10 @@ module _ ( obj-map-Precategory C D F x) ( obj-map-Precategory C D G x) - coherence-square-hom-family-map-Precategory : + naturality-hom-family-map-Precategory : hom-family-map-Precategory → {x y : obj-Precategory C} (f : hom-Precategory C x y) → UU l4 - coherence-square-hom-family-map-Precategory γ {x} {y} f = + naturality-hom-family-map-Precategory γ {x} {y} f = coherence-square-hom-Precategory D ( hom-map-Precategory C D F f) ( γ x) @@ -69,7 +69,7 @@ module _ hom-family-map-Precategory → UU (l1 ⊔ l2 ⊔ l4) is-natural-transformation-map-Precategory γ = {x y : obj-Precategory C} (f : hom-Precategory C x y) → - coherence-square-hom-family-map-Precategory γ f + naturality-hom-family-map-Precategory γ f natural-transformation-map-Precategory : UU (l1 ⊔ l2 ⊔ l4) natural-transformation-map-Precategory = @@ -80,11 +80,11 @@ module _ natural-transformation-map-Precategory → hom-family-map-Precategory hom-family-natural-transformation-map-Precategory = pr1 - coherence-square-natural-transformation-map-Precategory : + naturality-natural-transformation-map-Precategory : (γ : natural-transformation-map-Precategory) → is-natural-transformation-map-Precategory ( hom-family-natural-transformation-map-Precategory γ) - coherence-square-natural-transformation-map-Precategory = pr2 + naturality-natural-transformation-map-Precategory = pr2 ``` ## Composition and identity of natural transformations @@ -124,7 +124,7 @@ module _ ( ap ( comp-hom-Precategory' D ( hom-family-natural-transformation-map-Precategory C D F G α X)) - ( coherence-square-natural-transformation-map-Precategory C D G H β f)) ∙ + ( naturality-natural-transformation-map-Precategory C D G H β f)) ∙ ( associative-comp-hom-Precategory D ( hom-family-natural-transformation-map-Precategory C D G H β Y) ( hom-map-Precategory C D G f) @@ -132,7 +132,7 @@ module _ ( ap ( comp-hom-Precategory D ( hom-family-natural-transformation-map-Precategory C D G H β Y)) - ( coherence-square-natural-transformation-map-Precategory C D F G α f)) ∙ + ( naturality-natural-transformation-map-Precategory C D F G α f)) ∙ ( inv ( associative-comp-hom-Precategory D ( hom-family-natural-transformation-map-Precategory C D G H β Y) diff --git a/src/category-theory/nonunital-precategories.lagda.md b/src/category-theory/nonunital-precategories.lagda.md index 57edb55811..f758d636a4 100644 --- a/src/category-theory/nonunital-precategories.lagda.md +++ b/src/category-theory/nonunital-precategories.lagda.md @@ -14,6 +14,8 @@ open import foundation.dependent-pair-types open import foundation.identity-types open import foundation.propositions open import foundation.sets +open import foundation.truncated-types +open import foundation.truncation-levels open import foundation.universe-levels ``` @@ -169,6 +171,39 @@ module _ ( comp-hom-Nonunital-Precategory C) ``` +## Properties + +### If the objects of a nonunital precategory are `k`-truncated for non-negative `k`, the total hom-type is `k`-truncated + +```agda +module _ + {l1 l2 : Level} {k : 𝕋} (C : Nonunital-Precategory l1 l2) + where + + is-trunc-total-hom-is-trunc-obj-Nonunital-Precategory : + is-trunc (succ-𝕋 (succ-𝕋 k)) (obj-Nonunital-Precategory C) → + is-trunc (succ-𝕋 (succ-𝕋 k)) (total-hom-Nonunital-Precategory C) + is-trunc-total-hom-is-trunc-obj-Nonunital-Precategory is-trunc-obj-C = + is-trunc-Σ + ( is-trunc-obj-C) + ( λ x → + is-trunc-Σ + ( is-trunc-obj-C) + ( λ y → is-trunc-is-set k (is-set-hom-Nonunital-Precategory C x y))) + + total-hom-truncated-type-is-trunc-obj-Nonunital-Precategory : + is-trunc (succ-𝕋 (succ-𝕋 k)) (obj-Nonunital-Precategory C) → + Truncated-Type (l1 ⊔ l2) (succ-𝕋 (succ-𝕋 k)) + pr1 + ( total-hom-truncated-type-is-trunc-obj-Nonunital-Precategory + is-trunc-obj-C) = + total-hom-Nonunital-Precategory C + pr2 + ( total-hom-truncated-type-is-trunc-obj-Nonunital-Precategory + is-trunc-obj-C) = + is-trunc-total-hom-is-trunc-obj-Nonunital-Precategory is-trunc-obj-C +``` + ## Comments As discussed in [Semicategories](https://ncatlab.org/nlab/show/semicategory) at diff --git a/src/category-theory/opposite-categories.lagda.md b/src/category-theory/opposite-categories.lagda.md new file mode 100644 index 0000000000..b6b18e96af --- /dev/null +++ b/src/category-theory/opposite-categories.lagda.md @@ -0,0 +1,159 @@ +# Opposite categories + +```agda +module category-theory.opposite-categories where +``` + +
Imports + +```agda +open import category-theory.categories +open import category-theory.isomorphisms-in-precategories +open import category-theory.opposite-precategories +open import category-theory.precategories + +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.identity-types +open import foundation.involutions +open import foundation.sets +open import foundation.subtypes +open import foundation.transport-along-identifications +open import foundation.universe-levels +``` + +
+ +## Idea + +Let `C` be a [category](category-theory.categories.md), its **opposite +category** `Cᵒᵖ` is given by reversing every morphism. + +## Lemma + +### The underlying precategory is a category if and only if the opposite is a category + +```agda +abstract + is-category-opposite-is-category-Precategory : + {l1 l2 : Level} (C : Precategory l1 l2) → + is-category-Precategory C → + is-category-Precategory (opposite-Precategory C) + is-category-opposite-is-category-Precategory C is-category-C x y = + is-equiv-htpy-equiv + ( compute-iso-opposite-Precategory C ∘e + equiv-inv-iso-Precategory C ∘e + (_ , is-category-C x y)) + ( λ where + refl → + eq-type-subtype + ( is-iso-prop-Precategory (opposite-Precategory C)) + ( refl)) + +abstract + is-category-is-category-opposite-Precategory : + {l1 l2 : Level} (C : Precategory l1 l2) → + is-category-Precategory (opposite-Precategory C) → + is-category-Precategory C + is-category-is-category-opposite-Precategory C is-category-op-C = + tr + ( is-category-Precategory) + ( is-involution-opposite-Precategory C) + ( is-category-opposite-is-category-Precategory + ( opposite-Precategory C) + ( is-category-op-C)) +``` + +## Definitions + +### The opposite category + +```agda +module _ + {l1 l2 : Level} (C : Category l1 l2) + where + + obj-opposite-Category : UU l1 + obj-opposite-Category = obj-opposite-Precategory (precategory-Category C) + + hom-set-opposite-Category : (x y : obj-opposite-Category) → Set l2 + hom-set-opposite-Category = + hom-set-opposite-Precategory (precategory-Category C) + + hom-opposite-Category : (x y : obj-opposite-Category) → UU l2 + hom-opposite-Category = hom-opposite-Precategory (precategory-Category C) + + comp-hom-opposite-Category : + {x y z : obj-opposite-Category} → + hom-opposite-Category y z → + hom-opposite-Category x y → + hom-opposite-Category x z + comp-hom-opposite-Category = + comp-hom-opposite-Precategory (precategory-Category C) + + associative-comp-hom-opposite-Category : + {x y z w : obj-opposite-Category} + (h : hom-opposite-Category z w) + (g : hom-opposite-Category y z) + (f : hom-opposite-Category x y) → + ( comp-hom-opposite-Category (comp-hom-opposite-Category h g) f) ＝ + ( comp-hom-opposite-Category h (comp-hom-opposite-Category g f)) + associative-comp-hom-opposite-Category = + associative-comp-hom-opposite-Precategory (precategory-Category C) + + id-hom-opposite-Category : + {x : obj-opposite-Category} → hom-opposite-Category x x + id-hom-opposite-Category = + id-hom-opposite-Precategory (precategory-Category C) + + left-unit-law-comp-hom-opposite-Category : + {x y : obj-opposite-Category} + (f : hom-opposite-Category x y) → + comp-hom-opposite-Category id-hom-opposite-Category f ＝ f + left-unit-law-comp-hom-opposite-Category = + left-unit-law-comp-hom-opposite-Precategory (precategory-Category C) + + right-unit-law-comp-hom-opposite-Category : + {x y : obj-opposite-Category} (f : hom-opposite-Category x y) → + comp-hom-opposite-Category f id-hom-opposite-Category ＝ f + right-unit-law-comp-hom-opposite-Category = + right-unit-law-comp-hom-opposite-Precategory (precategory-Category C) + + precategory-opposite-Category : Precategory l1 l2 + precategory-opposite-Category = opposite-Precategory (precategory-Category C) + + opposite-Category : Category l1 l2 + pr1 opposite-Category = precategory-opposite-Category + pr2 opposite-Category = + is-category-opposite-is-category-Precategory + ( precategory-Category C) + ( is-category-Category C) +``` + +## Properties + +### The opposite category construction is an involution on the type of categories + +```agda +is-involution-opposite-Category : + {l1 l2 : Level} → is-involution (opposite-Category {l1} {l2}) +is-involution-opposite-Category C = + eq-type-subtype + ( is-category-prop-Precategory) + ( is-involution-opposite-Precategory (precategory-Category C)) + +involution-opposite-Category : + (l1 l2 : Level) → involution (Category l1 l2) +pr1 (involution-opposite-Category l1 l2) = opposite-Category +pr2 (involution-opposite-Category l1 l2) = is-involution-opposite-Category + +is-equiv-opposite-Category : + {l1 l2 : Level} → is-equiv (opposite-Category {l1} {l2}) +is-equiv-opposite-Category = + is-equiv-is-involution is-involution-opposite-Category + +equiv-opposite-Category : + (l1 l2 : Level) → Category l1 l2 ≃ Category l1 l2 +equiv-opposite-Category l1 l2 = + equiv-involution (involution-opposite-Category l1 l2) +``` diff --git a/src/category-theory/opposite-precategories.lagda.md b/src/category-theory/opposite-precategories.lagda.md index 6c1304e2e5..7f5bb7e217 100644 --- a/src/category-theory/opposite-precategories.lagda.md +++ b/src/category-theory/opposite-precategories.lagda.md @@ -7,10 +7,18 @@ module category-theory.opposite-precategories where
Imports ```agda +open import category-theory.composition-operations-on-binary-families-of-sets +open import category-theory.isomorphisms-in-precategories open import category-theory.precategories open import foundation.dependent-pair-types +open import foundation.equality-dependent-pair-types +open import foundation.equivalences +open import foundation.homotopies open import foundation.identity-types +open import foundation.involutions +open import foundation.propositions +open import foundation.sets open import foundation.universe-levels ``` @@ -23,20 +31,148 @@ precategory** `Cᵒᵖ` is given by reversing every morphism. ## Definition +### The opposite precategory + ```agda module _ {l1 l2 : Level} (C : Precategory l1 l2) where + obj-opposite-Precategory : UU l1 + obj-opposite-Precategory = obj-Precategory C + + hom-set-opposite-Precategory : (x y : obj-opposite-Precategory) → Set l2 + hom-set-opposite-Precategory x y = hom-set-Precategory C y x + + hom-opposite-Precategory : (x y : obj-opposite-Precategory) → UU l2 + hom-opposite-Precategory x y = type-Set (hom-set-opposite-Precategory x y) + + comp-hom-opposite-Precategory : + {x y z : obj-opposite-Precategory} → + hom-opposite-Precategory y z → + hom-opposite-Precategory x y → + hom-opposite-Precategory x z + comp-hom-opposite-Precategory g f = comp-hom-Precategory C f g + + associative-comp-hom-opposite-Precategory : + {x y z w : obj-opposite-Precategory} + (h : hom-opposite-Precategory z w) + (g : hom-opposite-Precategory y z) + (f : hom-opposite-Precategory x y) → + ( comp-hom-opposite-Precategory (comp-hom-opposite-Precategory h g) f) ＝ + ( comp-hom-opposite-Precategory h (comp-hom-opposite-Precategory g f)) + associative-comp-hom-opposite-Precategory h g f = + inv (associative-comp-hom-Precategory C f g h) + + id-hom-opposite-Precategory : + {x : obj-opposite-Precategory} → hom-opposite-Precategory x x + id-hom-opposite-Precategory = id-hom-Precategory C + + left-unit-law-comp-hom-opposite-Precategory : + {x y : obj-opposite-Precategory} + (f : hom-opposite-Precategory x y) → + comp-hom-opposite-Precategory id-hom-opposite-Precategory f ＝ f + left-unit-law-comp-hom-opposite-Precategory = + right-unit-law-comp-hom-Precategory C + + right-unit-law-comp-hom-opposite-Precategory : + {x y : obj-opposite-Precategory} (f : hom-opposite-Precategory x y) → + comp-hom-opposite-Precategory f id-hom-opposite-Precategory ＝ f + right-unit-law-comp-hom-opposite-Precategory = + left-unit-law-comp-hom-Precategory C + opposite-Precategory : Precategory l1 l2 - pr1 opposite-Precategory = obj-Precategory C - pr1 (pr2 opposite-Precategory) x y = hom-set-Precategory C y x - pr1 (pr1 (pr2 (pr2 opposite-Precategory))) f g = comp-hom-Precategory C g f - pr2 (pr1 (pr2 (pr2 opposite-Precategory))) f g h = - inv (associative-comp-hom-Precategory C h g f) - pr1 (pr2 (pr2 (pr2 opposite-Precategory))) x = id-hom-Precategory C {x} - pr1 (pr2 (pr2 (pr2 (pr2 opposite-Precategory)))) f = - right-unit-law-comp-hom-Precategory C f - pr2 (pr2 (pr2 (pr2 (pr2 opposite-Precategory)))) f = - left-unit-law-comp-hom-Precategory C f + pr1 opposite-Precategory = obj-opposite-Precategory + pr1 (pr2 opposite-Precategory) = hom-set-opposite-Precategory + pr1 (pr1 (pr2 (pr2 opposite-Precategory))) = comp-hom-opposite-Precategory + pr2 (pr1 (pr2 (pr2 opposite-Precategory))) = + associative-comp-hom-opposite-Precategory + pr1 (pr2 (pr2 (pr2 opposite-Precategory))) x = id-hom-opposite-Precategory {x} + pr1 (pr2 (pr2 (pr2 (pr2 opposite-Precategory)))) = + left-unit-law-comp-hom-opposite-Precategory + pr2 (pr2 (pr2 (pr2 (pr2 opposite-Precategory)))) = + right-unit-law-comp-hom-opposite-Precategory +``` + +## Properties + +### The opposite precategory construction is an involution on the type of precategories + +```agda +is-involution-opposite-Precategory : + {l1 l2 : Level} → is-involution (opposite-Precategory {l1} {l2}) +is-involution-opposite-Precategory C = + eq-pair-eq-pr2 + ( eq-pair-eq-pr2 + ( eq-pair-Σ + ( eq-pair-eq-pr2 + ( eq-is-prop + ( is-prop-is-associative-composition-operation-binary-family-Set + ( hom-set-Precategory C) + ( comp-hom-Precategory C)))) + ( eq-is-prop + ( is-prop-is-unital-composition-operation-binary-family-Set + ( hom-set-Precategory C) + ( comp-hom-Precategory C))))) + +involution-opposite-Precategory : + (l1 l2 : Level) → involution (Precategory l1 l2) +pr1 (involution-opposite-Precategory l1 l2) = opposite-Precategory +pr2 (involution-opposite-Precategory l1 l2) = is-involution-opposite-Precategory + +is-equiv-opposite-Precategory : + {l1 l2 : Level} → is-equiv (opposite-Precategory {l1} {l2}) +is-equiv-opposite-Precategory = + is-equiv-is-involution is-involution-opposite-Precategory + +equiv-opposite-Precategory : + (l1 l2 : Level) → Precategory l1 l2 ≃ Precategory l1 l2 +equiv-opposite-Precategory l1 l2 = + equiv-involution (involution-opposite-Precategory l1 l2) +``` + +### Computing the isomorphism sets of the opposite precategory + +```agda +module _ + {l1 l2 : Level} (C : Precategory l1 l2) {x y : obj-Precategory C} + where + + map-compute-iso-opposite-Precategory : + iso-Precategory C x y → iso-Precategory (opposite-Precategory C) y x + pr1 (map-compute-iso-opposite-Precategory f) = + hom-iso-Precategory C f + pr1 (pr2 (map-compute-iso-opposite-Precategory f)) = + hom-inv-iso-Precategory C f + pr1 (pr2 (pr2 (map-compute-iso-opposite-Precategory f))) = + is-retraction-hom-inv-iso-Precategory C f + pr2 (pr2 (pr2 (map-compute-iso-opposite-Precategory f))) = + is-section-hom-inv-iso-Precategory C f + + map-inv-compute-iso-opposite-Precategory : + iso-Precategory (opposite-Precategory C) y x → iso-Precategory C x y + pr1 (map-inv-compute-iso-opposite-Precategory f) = + hom-iso-Precategory (opposite-Precategory C) f + pr1 (pr2 (map-inv-compute-iso-opposite-Precategory f)) = + hom-inv-iso-Precategory (opposite-Precategory C) f + pr1 (pr2 (pr2 (map-inv-compute-iso-opposite-Precategory f))) = + is-retraction-hom-inv-iso-Precategory (opposite-Precategory C) f + pr2 (pr2 (pr2 (map-inv-compute-iso-opposite-Precategory f))) = + is-section-hom-inv-iso-Precategory (opposite-Precategory C) f + + is-equiv-map-compute-iso-opposite-Precategory : + is-equiv (map-compute-iso-opposite-Precategory) + pr1 (pr1 is-equiv-map-compute-iso-opposite-Precategory) = + map-inv-compute-iso-opposite-Precategory + pr2 (pr1 is-equiv-map-compute-iso-opposite-Precategory) = refl-htpy + pr1 (pr2 is-equiv-map-compute-iso-opposite-Precategory) = + map-inv-compute-iso-opposite-Precategory + pr2 (pr2 is-equiv-map-compute-iso-opposite-Precategory) = refl-htpy + + compute-iso-opposite-Precategory : + iso-Precategory C x y ≃ iso-Precategory (opposite-Precategory C) y x + pr1 compute-iso-opposite-Precategory = + map-compute-iso-opposite-Precategory + pr2 compute-iso-opposite-Precategory = + is-equiv-map-compute-iso-opposite-Precategory ``` diff --git a/src/category-theory/opposite-preunivalent-categories.lagda.md b/src/category-theory/opposite-preunivalent-categories.lagda.md new file mode 100644 index 0000000000..e599c994fb --- /dev/null +++ b/src/category-theory/opposite-preunivalent-categories.lagda.md @@ -0,0 +1,182 @@ +# Opposite preunivalent categories + +```agda +module category-theory.opposite-preunivalent-categories where +``` + +
Imports + +```agda +open import category-theory.isomorphisms-in-precategories +open import category-theory.opposite-precategories +open import category-theory.precategories +open import category-theory.preunivalent-categories + +open import foundation.dependent-pair-types +open import foundation.embeddings +open import foundation.equivalences +open import foundation.identity-types +open import foundation.involutions +open import foundation.sets +open import foundation.subtypes +open import foundation.transport-along-identifications +open import foundation.universe-levels +``` + +
+ +## Idea + +Let `C` be a +[preunivalent category](category-theory.preunivalent-categories.md), its +**opposite preunivalent category** `Cᵒᵖ` is given by reversing every morphism. + +## Lemma + +### A precategory is preunivalent if and only if the opposite is preunivalent + +```agda +abstract + is-preunivalent-opposite-is-preunivalent-Precategory : + {l1 l2 : Level} (C : Precategory l1 l2) → + is-preunivalent-Precategory C → + is-preunivalent-Precategory (opposite-Precategory C) + is-preunivalent-opposite-is-preunivalent-Precategory C is-preunivalent-C x y = + is-emb-htpy-emb + ( comp-emb + ( emb-equiv + ( compute-iso-opposite-Precategory C ∘e equiv-inv-iso-Precategory C)) + ( _ , is-preunivalent-C x y)) + ( λ where + refl → + eq-type-subtype + ( is-iso-prop-Precategory (opposite-Precategory C)) + ( refl)) + +abstract + is-preunivalent-is-preunivalent-opposite-Precategory : + {l1 l2 : Level} (C : Precategory l1 l2) → + is-preunivalent-Precategory (opposite-Precategory C) → + is-preunivalent-Precategory C + is-preunivalent-is-preunivalent-opposite-Precategory C is-preunivalent-op-C = + tr + ( is-preunivalent-Precategory) + ( is-involution-opposite-Precategory C) + ( is-preunivalent-opposite-is-preunivalent-Precategory + ( opposite-Precategory C) + ( is-preunivalent-op-C)) +``` + +## Definitions + +### The opposite preunivalent category + +```agda +module _ + {l1 l2 : Level} (C : Preunivalent-Category l1 l2) + where + + obj-opposite-Preunivalent-Category : UU l1 + obj-opposite-Preunivalent-Category = + obj-opposite-Precategory (precategory-Preunivalent-Category C) + + hom-set-opposite-Preunivalent-Category : + (x y : obj-opposite-Preunivalent-Category) → Set l2 + hom-set-opposite-Preunivalent-Category = + hom-set-opposite-Precategory (precategory-Preunivalent-Category C) + + hom-opposite-Preunivalent-Category : + (x y : obj-opposite-Preunivalent-Category) → UU l2 + hom-opposite-Preunivalent-Category = + hom-opposite-Precategory (precategory-Preunivalent-Category C) + + comp-hom-opposite-Preunivalent-Category : + {x y z : obj-opposite-Preunivalent-Category} → + hom-opposite-Preunivalent-Category y z → + hom-opposite-Preunivalent-Category x y → + hom-opposite-Preunivalent-Category x z + comp-hom-opposite-Preunivalent-Category = + comp-hom-opposite-Precategory (precategory-Preunivalent-Category C) + + associative-comp-hom-opposite-Preunivalent-Category : + {x y z w : obj-opposite-Preunivalent-Category} + (h : hom-opposite-Preunivalent-Category z w) + (g : hom-opposite-Preunivalent-Category y z) + (f : hom-opposite-Preunivalent-Category x y) → + ( comp-hom-opposite-Preunivalent-Category + ( comp-hom-opposite-Preunivalent-Category h g) (f)) ＝ + ( comp-hom-opposite-Preunivalent-Category + ( h) (comp-hom-opposite-Preunivalent-Category g f)) + associative-comp-hom-opposite-Preunivalent-Category = + associative-comp-hom-opposite-Precategory + ( precategory-Preunivalent-Category C) + + id-hom-opposite-Preunivalent-Category : + {x : obj-opposite-Preunivalent-Category} → + hom-opposite-Preunivalent-Category x x + id-hom-opposite-Preunivalent-Category = + id-hom-opposite-Precategory (precategory-Preunivalent-Category C) + + left-unit-law-comp-hom-opposite-Preunivalent-Category : + {x y : obj-opposite-Preunivalent-Category} + (f : hom-opposite-Preunivalent-Category x y) → + comp-hom-opposite-Preunivalent-Category + ( id-hom-opposite-Preunivalent-Category) + ( f) ＝ + f + left-unit-law-comp-hom-opposite-Preunivalent-Category = + left-unit-law-comp-hom-opposite-Precategory + ( precategory-Preunivalent-Category C) + + right-unit-law-comp-hom-opposite-Preunivalent-Category : + {x y : obj-opposite-Preunivalent-Category} + (f : hom-opposite-Preunivalent-Category x y) → + comp-hom-opposite-Preunivalent-Category + ( f) (id-hom-opposite-Preunivalent-Category) ＝ + ( f) + right-unit-law-comp-hom-opposite-Preunivalent-Category = + right-unit-law-comp-hom-opposite-Precategory + ( precategory-Preunivalent-Category C) + + precategory-opposite-Preunivalent-Category : Precategory l1 l2 + precategory-opposite-Preunivalent-Category = + opposite-Precategory (precategory-Preunivalent-Category C) + + opposite-Preunivalent-Category : Preunivalent-Category l1 l2 + pr1 opposite-Preunivalent-Category = + precategory-opposite-Preunivalent-Category + pr2 opposite-Preunivalent-Category = + is-preunivalent-opposite-is-preunivalent-Precategory + ( precategory-Preunivalent-Category C) + ( is-preunivalent-Preunivalent-Category C) +``` + +## Properties + +### The opposite preunivalent category construction is an involution on the type of preunivalent categories + +```agda +is-involution-opposite-Preunivalent-Category : + {l1 l2 : Level} → is-involution (opposite-Preunivalent-Category {l1} {l2}) +is-involution-opposite-Preunivalent-Category C = + eq-type-subtype + ( is-preunivalent-prop-Precategory) + ( is-involution-opposite-Precategory (precategory-Preunivalent-Category C)) + +involution-opposite-Preunivalent-Category : + (l1 l2 : Level) → involution (Preunivalent-Category l1 l2) +pr1 (involution-opposite-Preunivalent-Category l1 l2) = + opposite-Preunivalent-Category +pr2 (involution-opposite-Preunivalent-Category l1 l2) = + is-involution-opposite-Preunivalent-Category + +is-equiv-opposite-Preunivalent-Category : + {l1 l2 : Level} → is-equiv (opposite-Preunivalent-Category {l1} {l2}) +is-equiv-opposite-Preunivalent-Category = + is-equiv-is-involution is-involution-opposite-Preunivalent-Category + +equiv-opposite-Preunivalent-Category : + (l1 l2 : Level) → Preunivalent-Category l1 l2 ≃ Preunivalent-Category l1 l2 +equiv-opposite-Preunivalent-Category l1 l2 = + equiv-involution (involution-opposite-Preunivalent-Category l1 l2) +``` diff --git a/src/category-theory/precategories.lagda.md b/src/category-theory/precategories.lagda.md index eee5a8b052..1df30c30d4 100644 --- a/src/category-theory/precategories.lagda.md +++ b/src/category-theory/precategories.lagda.md @@ -16,6 +16,8 @@ open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets +open import foundation.truncated-types +open import foundation.truncation-levels open import foundation.universe-levels ``` @@ -167,6 +169,21 @@ module _ associative-comp-hom-Precategory C ``` +### The total hom-type of a precategory + +```agda +total-hom-Precategory : + {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2) +total-hom-Precategory C = + total-hom-Nonunital-Precategory (nonunital-precategory-Precategory C) + +obj-total-hom-Precategory : + {l1 l2 : Level} (C : Precategory l1 l2) → + total-hom-Precategory C → obj-Precategory C × obj-Precategory C +obj-total-hom-Precategory C = + obj-total-hom-Nonunital-Precategory (nonunital-precategory-Precategory C) +``` + ### Equalities induce morphisms ```agda @@ -184,21 +201,6 @@ module _ hom-inv-eq-Precategory x y = hom-eq-Precategory y x ∘ inv ``` -### The total hom-type of a precategory - -```agda -total-hom-Precategory : - {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2) -total-hom-Precategory C = - total-hom-Nonunital-Precategory (nonunital-precategory-Precategory C) - -obj-total-hom-Precategory : - {l1 l2 : Level} (C : Precategory l1 l2) → - total-hom-Precategory C → obj-Precategory C × obj-Precategory C -obj-total-hom-Precategory C = - obj-total-hom-Nonunital-Precategory (nonunital-precategory-Precategory C) -``` - ### Pre- and postcomposition by a morphism ```agda @@ -214,3 +216,25 @@ postcomp-hom-Precategory : hom-Precategory C z x → hom-Precategory C z y postcomp-hom-Precategory C f z = comp-hom-Precategory C f ``` + +## If the objects of a precategory are `k`-truncated for non-negative `k`, the total hom-type is `k`-truncated + +```agda +module _ + {l1 l2 : Level} {k : 𝕋} (C : Precategory l1 l2) + where + + is-trunc-total-hom-is-trunc-obj-Precategory : + is-trunc (succ-𝕋 (succ-𝕋 k)) (obj-Precategory C) → + is-trunc (succ-𝕋 (succ-𝕋 k)) (total-hom-Precategory C) + is-trunc-total-hom-is-trunc-obj-Precategory = + is-trunc-total-hom-is-trunc-obj-Nonunital-Precategory + ( nonunital-precategory-Precategory C) + + total-hom-truncated-type-is-trunc-obj-Precategory : + is-trunc (succ-𝕋 (succ-𝕋 k)) (obj-Precategory C) → + Truncated-Type (l1 ⊔ l2) (succ-𝕋 (succ-𝕋 k)) + total-hom-truncated-type-is-trunc-obj-Precategory = + total-hom-truncated-type-is-trunc-obj-Nonunital-Precategory + ( nonunital-precategory-Precategory C) +``` diff --git a/src/category-theory/presheaf-categories.lagda.md b/src/category-theory/presheaf-categories.lagda.md index f5e7ddd193..abf3a474e7 100644 --- a/src/category-theory/presheaf-categories.lagda.md +++ b/src/category-theory/presheaf-categories.lagda.md @@ -7,14 +7,13 @@ module category-theory.presheaf-categories where
Imports ```agda -open import category-theory.category-of-functors-from-small-to-large-categories +open import category-theory.categories +open import category-theory.copresheaf-categories open import category-theory.large-categories open import category-theory.large-precategories open import category-theory.opposite-precategories open import category-theory.precategories -open import category-theory.precategory-of-functors-from-small-to-large-precategories -open import foundation.category-of-sets open import foundation.universe-levels ``` @@ -22,63 +21,140 @@ open import foundation.universe-levels ## Idea -Given any [precategory](category-theory.precategories.md) `C`, we can form its -**presheaf category** as the -[large category](category-theory.large-categories.md) of -[functors](category-theory.functors-from-small-to-large-precategories.md) from -the [opposite precategory](category-theory.opposite-precategories.md) of `C`, +Given a [precategory](category-theory.precategories.md) `C`, we can form its +**presheaf [category](category-theory.large-categories.md)** as the +[large category of functors](category-theory.functors-from-small-to-large-precategories.md) +from the [opposite of](category-theory.opposite-precategories.md) `C`, `Cᵒᵖ`, into the [large category of sets](foundation.category-of-sets.md) ```text Cᵒᵖ → Set. ``` -Dually, we can form the **copresheaf category** of `C` by taking the large -functor category - -```text - C → Set. -``` +To this large category, there is an associated +[small category](category-theory.categories.md) of small presheaves, taking +values in small [sets](foundation-core.sets.md). ## Definitions -### The copresheaf category of a precategory +### The large category of presheaves on a precategory ```agda module _ {l1 l2 : Level} (C : Precategory l1 l2) where - copresheaf-Large-Precategory : + presheaf-Large-Precategory : Large-Precategory (λ l → l1 ⊔ l2 ⊔ lsuc l) (λ l l' → l1 ⊔ l2 ⊔ l ⊔ l') - copresheaf-Large-Precategory = - functor-large-precategory-Small-Large-Precategory C Set-Large-Precategory + presheaf-Large-Precategory = + copresheaf-Large-Precategory (opposite-Precategory C) + + is-large-category-presheaf-Large-Category : + is-large-category-Large-Precategory presheaf-Large-Precategory + is-large-category-presheaf-Large-Category = + is-large-category-copresheaf-Large-Category (opposite-Precategory C) - copresheaf-Large-Category : + presheaf-Large-Category : Large-Category (λ l → l1 ⊔ l2 ⊔ lsuc l) (λ l l' → l1 ⊔ l2 ⊔ l ⊔ l') - large-precategory-Large-Category copresheaf-Large-Category = - copresheaf-Large-Precategory - is-large-category-Large-Category copresheaf-Large-Category = - is-large-category-functor-large-precategory-is-large-category-Small-Large-Precategory - ( C) - ( Set-Large-Precategory) - ( is-large-category-Set-Large-Precategory) + presheaf-Large-Category = copresheaf-Large-Category (opposite-Precategory C) ``` -### The presheaf category of a precategory +We record the components of the large category of presheaves on a precategory. + +```agda + obj-presheaf-Large-Category = + obj-Large-Precategory presheaf-Large-Precategory + + hom-set-presheaf-Large-Category = + hom-set-Large-Precategory presheaf-Large-Precategory + + hom-presheaf-Large-Category = + hom-Large-Precategory presheaf-Large-Precategory + + comp-hom-presheaf-Large-Category = + comp-hom-Large-Precategory presheaf-Large-Precategory + + id-hom-presheaf-Large-Category = + id-hom-Large-Precategory presheaf-Large-Precategory + + associative-comp-hom-presheaf-Large-Category = + associative-comp-hom-Large-Precategory presheaf-Large-Precategory + + left-unit-law-comp-hom-presheaf-Large-Category = + left-unit-law-comp-hom-Large-Precategory presheaf-Large-Precategory + + right-unit-law-comp-hom-presheaf-Large-Category = + right-unit-law-comp-hom-Large-Precategory presheaf-Large-Precategory +``` + +### The category of small presheaves on a precategory ```agda module _ - {l1 l2 : Level} (C : Precategory l1 l2) + {l1 l2 : Level} (C : Precategory l1 l2) (l : Level) where - presheaf-Large-Precategory : - Large-Precategory (λ l → l1 ⊔ l2 ⊔ lsuc l) (λ l l' → l1 ⊔ l2 ⊔ l ⊔ l') - presheaf-Large-Precategory = - copresheaf-Large-Precategory (opposite-Precategory C) + presheaf-Precategory : Precategory (l1 ⊔ l2 ⊔ lsuc l) (l1 ⊔ l2 ⊔ l) + presheaf-Precategory = + precategory-Large-Precategory (presheaf-Large-Precategory C) l - presheaf-Large-Category : - Large-Category (λ l → l1 ⊔ l2 ⊔ lsuc l) (λ l l' → l1 ⊔ l2 ⊔ l ⊔ l') - presheaf-Large-Category = - copresheaf-Large-Category (opposite-Precategory C) + presheaf-Category : Category (l1 ⊔ l2 ⊔ lsuc l) (l1 ⊔ l2 ⊔ l) + presheaf-Category = category-Large-Category (presheaf-Large-Category C) l +``` + +We also record the components of the category of small presheaves on a +precategory. + +```agda + obj-presheaf-Category = + obj-Precategory presheaf-Precategory + + hom-set-presheaf-Category = + hom-set-Precategory presheaf-Precategory + + hom-presheaf-Category = + hom-Precategory presheaf-Precategory + + comp-hom-presheaf-Category = + comp-hom-Precategory presheaf-Precategory + + id-hom-presheaf-Category = + id-hom-Precategory presheaf-Precategory + + associative-comp-hom-presheaf-Category = + associative-comp-hom-Precategory presheaf-Precategory + + left-unit-law-comp-hom-presheaf-Category = + left-unit-law-comp-hom-Precategory presheaf-Precategory + + right-unit-law-comp-hom-presheaf-Category = + right-unit-law-comp-hom-Precategory presheaf-Precategory ``` + +### Sections of presheaves + +As a choice of universe level must be made to talk about sections of presheaves, +this notion coincides for the large and small category of presheaves. + +```agda +module _ + {l1 l2 l3 : Level} (C : Precategory l1 l2) + where + + section-presheaf-Category : + (F : obj-presheaf-Category C l3) (c : obj-Precategory C) → UU l3 + section-presheaf-Category = + section-copresheaf-Category (opposite-Precategory C) +``` + +## See also + +- [The Yoneda lemma](category-theory.yoneda-lemma-precategories.md) + +## External links + +- [Presheaf precategories](https://1lab.dev/Cat.Functor.Base.html#presheaf-precategories) + at 1lab +- [category of presheaves](https://ncatlab.org/nlab/show/category+of+presheaves) + at nlab +- [presheaf](https://ncatlab.org/nlab/show/presheaf) at nlab diff --git a/src/category-theory/preunivalent-categories.lagda.md b/src/category-theory/preunivalent-categories.lagda.md index 79e238fea8..718ba007bd 100644 --- a/src/category-theory/preunivalent-categories.lagda.md +++ b/src/category-theory/preunivalent-categories.lagda.md @@ -12,6 +12,7 @@ open import category-theory.isomorphisms-in-precategories open import category-theory.precategories open import foundation.1-types +open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.identity-types @@ -39,11 +40,14 @@ preunivalent categories generalize both [strict categories](category-theory.strict-categories.md), which are precategories whose objects form a [set](foundation-core.sets.md). -Summarized, the preunivalence condition on precategories states that the type of -objects is a subgroupoid of the [groupoid](category-theory.groupoids.md) of -isomorphisms. For univalent categories the groupoid of objects is equivalent to -the groupoid of isomorphisms, while for strict categories the groupoid of -objects is discrete. +The preunivalence condition on precategories states that the type of objects is +a subgroupoid of the [groupoid](category-theory.groupoids.md) of isomorphisms. +For univalent categories the groupoid of objects is equivalent to the groupoid +of isomorphisms, while for strict categories the groupoid of objects is +discrete. Indeed, in this sense preunivalence provides a generalization of both +notions of "category", with _no more structure_. This is opposed to the even +more general notion of precategory, where the homotopy structure on the objects +can be almost completely unrelated to the homotopy structure of the morphisms. ## Definitions @@ -151,6 +155,30 @@ module _ is-preunivalent-Preunivalent-Category : is-preunivalent-Precategory precategory-Preunivalent-Category is-preunivalent-Preunivalent-Category = pr2 C + + emb-iso-eq-Preunivalent-Category : + {x y : obj-Preunivalent-Category} → + (x ＝ y) ↪ (iso-Precategory precategory-Preunivalent-Category x y) + pr1 (emb-iso-eq-Preunivalent-Category {x} {y}) = + iso-eq-Precategory precategory-Preunivalent-Category x y + pr2 (emb-iso-eq-Preunivalent-Category {x} {y}) = + is-preunivalent-Preunivalent-Category x y +``` + +### The total hom-type of a preunivalent category + +```agda +total-hom-Preunivalent-Category : + {l1 l2 : Level} (C : Preunivalent-Category l1 l2) → UU (l1 ⊔ l2) +total-hom-Preunivalent-Category C = + total-hom-Precategory (precategory-Preunivalent-Category C) + +obj-total-hom-Preunivalent-Category : + {l1 l2 : Level} (C : Preunivalent-Category l1 l2) → + total-hom-Preunivalent-Category C → + obj-Preunivalent-Category C × obj-Preunivalent-Category C +obj-total-hom-Preunivalent-Category C = + obj-total-hom-Precategory (precategory-Preunivalent-Category C) ``` ### Equalities induce morphisms @@ -223,6 +251,27 @@ module _ pr2 obj-1-type-Preunivalent-Category = is-1-type-obj-Preunivalent-Category ``` +### The total hom-type of a preunivalent category is a 1-type + +```agda +module _ + {l1 l2 : Level} (C : Preunivalent-Category l1 l2) + where + + is-1-type-total-hom-Preunivalent-Category : + is-1-type (total-hom-Preunivalent-Category C) + is-1-type-total-hom-Preunivalent-Category = + is-trunc-total-hom-is-trunc-obj-Precategory + ( precategory-Preunivalent-Category C) + ( is-1-type-obj-Preunivalent-Category C) + + total-hom-1-type-Preunivalent-Category : 1-Type (l1 ⊔ l2) + total-hom-1-type-Preunivalent-Category = + total-hom-truncated-type-is-trunc-obj-Precategory + ( precategory-Preunivalent-Category C) + ( is-1-type-obj-Preunivalent-Category C) +``` + ## See also - [The preunivalence axiom](foundation.preunivalence.md) diff --git a/src/category-theory/products-in-precategories.lagda.md b/src/category-theory/products-in-precategories.lagda.md index 9933dde857..cb10bb605d 100644 --- a/src/category-theory/products-in-precategories.lagda.md +++ b/src/category-theory/products-in-precategories.lagda.md @@ -14,6 +14,7 @@ open import foundation.cartesian-product-types open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.identity-types +open import foundation.iterated-dependent-product-types open import foundation.propositions open import foundation.unique-existence open import foundation.universe-levels @@ -126,7 +127,7 @@ module _ is-prop-is-product-Precategory : is-prop (is-product-Precategory C x y p l r) is-prop-is-product-Precategory = - is-prop-Π³ (λ z f g → is-property-is-contr) + is-prop-iterated-Π 3 (λ z f g → is-property-is-contr) is-product-prop-Precategory : Prop (l1 ⊔ l2) pr1 is-product-prop-Precategory = is-product-Precategory C x y p l r diff --git a/src/category-theory/pullbacks-in-precategories.lagda.md b/src/category-theory/pullbacks-in-precategories.lagda.md index 6a5d2efeb8..77bff5be49 100644 --- a/src/category-theory/pullbacks-in-precategories.lagda.md +++ b/src/category-theory/pullbacks-in-precategories.lagda.md @@ -14,6 +14,7 @@ open import foundation.cartesian-product-types open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.identity-types +open import foundation.iterated-dependent-product-types open import foundation.propositions open import foundation.unique-existence open import foundation.universe-levels @@ -159,7 +160,8 @@ module _ is-prop-is-pullback-Precategory : is-prop (is-pullback-Precategory C x y z f g w p₁ p₂ α) is-prop-is-pullback-Precategory = - is-prop-Π³ (λ w' p₁' p₂' → is-prop-function-type is-property-is-contr) + is-prop-iterated-Π 3 + ( λ w' p₁' p₂' → is-prop-function-type is-property-is-contr) is-pullback-prop-Precategory : Prop (l1 ⊔ l2) pr1 is-pullback-prop-Precategory = diff --git a/src/category-theory/replete-subprecategories.lagda.md b/src/category-theory/replete-subprecategories.lagda.md new file mode 100644 index 0000000000..f438e35093 --- /dev/null +++ b/src/category-theory/replete-subprecategories.lagda.md @@ -0,0 +1,332 @@ +# Replete subprecategories + +```agda +module category-theory.replete-subprecategories where +``` + +
Imports + +```agda +open import category-theory.categories +open import category-theory.isomorphism-induction-categories +open import category-theory.isomorphisms-in-precategories +open import category-theory.isomorphisms-in-subprecategories +open import category-theory.precategories +open import category-theory.subprecategories + +open import foundation.cartesian-product-types +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.functoriality-dependent-pair-types +open import foundation.iterated-dependent-product-types +open import foundation.propositions +open import foundation.subsingleton-induction +open import foundation.subtypes +open import foundation.transport-along-identifications +open import foundation.type-arithmetic-cartesian-product-types +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels +``` + +
+ +## Idea + +A **replete subprecategory** of a [precategory](category-theory.categories.md) +`C` is a [subprecategory](category-theory.subprecategories.md) `P` that is +closed under [isomorphisms](category-theory.isomorphisms-in-precategories.md): + +Given an object `x` in `P`, then every isomorphism `f : x ≅ y` in `C`, is +contained in `P`. + +## Definitions + +### The predicate on a subprecategory of being closed under isomorphic objects + +We can define what it means for subprecategories to have objects that are closed +under isomorphisms. Observe that this is not yet the correct definition of a +replete subprecategory. + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + where + + contains-iso-obj-Subprecategory : UU (l1 ⊔ l2 ⊔ l3) + contains-iso-obj-Subprecategory = + (x : obj-Subprecategory C P) (y : obj-Precategory C) → + iso-Precategory C (inclusion-obj-Subprecategory C P x) y → + is-in-obj-Subprecategory C P y + + is-prop-contains-iso-obj-Subprecategory : + is-prop contains-iso-obj-Subprecategory + is-prop-contains-iso-obj-Subprecategory = + is-prop-iterated-Π 3 (λ x y f → is-prop-is-in-obj-Subprecategory C P y) + + contains-iso-obj-prop-Subprecategory : Prop (l1 ⊔ l2 ⊔ l3) + pr1 contains-iso-obj-prop-Subprecategory = contains-iso-obj-Subprecategory + pr2 contains-iso-obj-prop-Subprecategory = + is-prop-contains-iso-obj-Subprecategory +``` + +### The predicate of being a replete subprecategory + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + where + + is-replete-Subprecategory : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) + is-replete-Subprecategory = + (x : obj-Subprecategory C P) + (y : obj-Precategory C) + (f : iso-Precategory C (inclusion-obj-Subprecategory C P x) y) → + Σ ( is-in-obj-Subprecategory C P y) + ( λ y₀ → is-in-iso-obj-subprecategory-Subprecategory C P {x} {y , y₀} f) + + is-prop-is-replete-Subprecategory : + is-prop is-replete-Subprecategory + is-prop-is-replete-Subprecategory = + is-prop-iterated-Π 3 + ( λ x y f → + is-prop-Σ + ( is-prop-is-in-obj-Subprecategory C P y) + ( λ _ → is-prop-is-in-iso-obj-subprecategory-Subprecategory C P f)) + + is-replete-prop-Subprecategory : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4) + pr1 is-replete-prop-Subprecategory = is-replete-Subprecategory + pr2 is-replete-prop-Subprecategory = + is-prop-is-replete-Subprecategory +``` + +### The type of replete subprecategories + +```agda +Replete-Subprecategory : + {l1 l2 : Level} (l3 l4 : Level) (C : Precategory l1 l2) → + UU (l1 ⊔ l2 ⊔ lsuc l3 ⊔ lsuc l4) +Replete-Subprecategory l3 l4 C = + Σ (Subprecategory l3 l4 C) (is-replete-Subprecategory C) + +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Replete-Subprecategory l3 l4 C) + where + + subprecategory-Replete-Subprecategory : Subprecategory l3 l4 C + subprecategory-Replete-Subprecategory = pr1 P + + is-replete-Replete-Subprecategory : + is-replete-Subprecategory C subprecategory-Replete-Subprecategory + is-replete-Replete-Subprecategory = pr2 P +``` + +## Properties + +### A slight reformulation of repleteness + +In our main definition of repleteness, the containment proof of the isomorphism +must be fixed at the left end-point. This is of course not necessary, so we can +ask for a slighty relaxed proof of repleteness: + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + where + + is-unfixed-replete-Subprecategory : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) + is-unfixed-replete-Subprecategory = + (x : obj-Subprecategory C P) + (y : obj-Precategory C) + (f : iso-Precategory C (inclusion-obj-Subprecategory C P x) y) → + is-in-iso-Subprecategory C P f + + is-prop-is-unfixed-replete-Subprecategory : + is-prop (is-unfixed-replete-Subprecategory) + is-prop-is-unfixed-replete-Subprecategory = + is-prop-iterated-Π 3 + ( λ x y f → is-prop-is-in-iso-Subprecategory C P f) + + is-unfixed-replete-prop-Subprecategory : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4) + pr1 is-unfixed-replete-prop-Subprecategory = + is-unfixed-replete-Subprecategory + pr2 is-unfixed-replete-prop-Subprecategory = + is-prop-is-unfixed-replete-Subprecategory + + is-unfixed-replete-is-replete-Subprecategory : + is-replete-Subprecategory C P → is-unfixed-replete-Subprecategory + pr1 (is-unfixed-replete-is-replete-Subprecategory replete' (x , x₀) y f) = x₀ + pr2 (is-unfixed-replete-is-replete-Subprecategory replete' x y f) = + replete' x y f + + is-replete-is-unfixed-replete-Subprecategory : + is-unfixed-replete-Subprecategory → is-replete-Subprecategory C P + is-replete-is-unfixed-replete-Subprecategory is-unfixed-replete-P x y f = + ind-subsingleton + ( is-prop-is-in-obj-Subprecategory C P (pr1 x)) + { λ x₀ → + Σ ( is-in-obj-Subprecategory C P y) + ( λ y₀ → + is-in-iso-obj-subprecategory-Subprecategory C P + { inclusion-obj-Subprecategory C P x , x₀} {y , y₀} f)} + ( pr1 (is-unfixed-replete-P x y f)) + ( pr2 (is-unfixed-replete-P x y f)) + ( is-in-obj-inclusion-obj-Subprecategory C P x) +``` + +### Isomorphism-sets in replete subprecategories are equivalent to isomorphism-sets in the base precategory + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + (is-replete-P : is-replete-Subprecategory C P) + {x y : obj-Subprecategory C P} (f : hom-Subprecategory C P x y) + where + + is-iso-is-iso-base-is-replete-Subprecategory : + is-iso-Precategory C (inclusion-hom-Subprecategory C P x y f) → + is-iso-Subprecategory C P f + pr1 (pr1 (is-iso-is-iso-base-is-replete-Subprecategory is-iso-C-f)) = + hom-inv-is-iso-Precategory C is-iso-C-f + pr2 (pr1 (is-iso-is-iso-base-is-replete-Subprecategory is-iso-C-f)) = + ind-subsingleton + ( is-prop-is-in-obj-Subprecategory C P (pr1 y)) + { λ y₀ → + is-in-hom-obj-subprecategory-Subprecategory C P (pr1 y , y₀) x + ( hom-inv-is-iso-Precategory C is-iso-C-f)} + ( pr1 (is-replete-P x (pr1 y) (pr1 f , is-iso-C-f))) + ( pr2 (pr2 (is-replete-P x (pr1 y) (pr1 f , is-iso-C-f)))) + ( pr2 y) + pr1 (pr2 (is-iso-is-iso-base-is-replete-Subprecategory is-iso-C-f)) = + eq-type-subtype + ( subtype-hom-obj-subprecategory-Subprecategory C P y y) + ( is-section-hom-inv-is-iso-Precategory C is-iso-C-f) + pr2 (pr2 (is-iso-is-iso-base-is-replete-Subprecategory is-iso-C-f)) = + eq-type-subtype + ( subtype-hom-obj-subprecategory-Subprecategory C P x x) + ( is-retraction-hom-inv-is-iso-Precategory C is-iso-C-f) + + is-equiv-is-iso-is-iso-base-is-replete-Subprecategory : + is-equiv is-iso-is-iso-base-is-replete-Subprecategory + is-equiv-is-iso-is-iso-base-is-replete-Subprecategory = + is-equiv-is-prop + ( is-prop-is-iso-Precategory C (inclusion-hom-Subprecategory C P x y f)) + ( is-prop-is-iso-Subprecategory C P f) + ( is-iso-base-is-iso-Subprecategory C P) + + equiv-is-iso-is-iso-base-is-replete-Subprecategory : + is-iso-Precategory C (inclusion-hom-Subprecategory C P x y f) ≃ + is-iso-Subprecategory C P f + pr1 equiv-is-iso-is-iso-base-is-replete-Subprecategory = + is-iso-is-iso-base-is-replete-Subprecategory + pr2 equiv-is-iso-is-iso-base-is-replete-Subprecategory = + is-equiv-is-iso-is-iso-base-is-replete-Subprecategory + +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + (is-replete-P : is-replete-Subprecategory C P) + (x y : obj-Subprecategory C P) + where + + compute-iso-is-replete-Subprecategory : + iso-Precategory C + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y) ≃ + iso-Subprecategory C P x y + compute-iso-is-replete-Subprecategory = + ( equiv-tot + ( equiv-is-iso-is-iso-base-is-replete-Subprecategory + C P is-replete-P {x} {y})) ∘e + ( inv-associative-Σ _ _ _) ∘e + ( equiv-tot + ( λ f → + ( commutative-prod) ∘e + ( inv-right-unit-law-Σ-is-contr + ( λ is-iso-C-f → is-proof-irrelevant-is-prop + ( is-prop-is-in-hom-obj-subprecategory-Subprecategory C P x y f) + ( ind-subsingleton + ( is-prop-is-in-obj-Subprecategory C P (pr1 y)) + { λ y₀ → + is-in-hom-obj-subprecategory-Subprecategory + C P x (pr1 y , y₀) f} + ( is-replete-P x (pr1 y) (f , is-iso-C-f) .pr1) + ( is-replete-P x (pr1 y) (f , is-iso-C-f) .pr2 .pr1) + ( pr2 y)))))) + + inv-compute-iso-is-replete-Subprecategory : + iso-Subprecategory C P x y ≃ + iso-Precategory C + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y) + inv-compute-iso-is-replete-Subprecategory = + inv-equiv compute-iso-is-replete-Subprecategory +``` + +### Subprecategories of categories are replete + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + (is-category-C : is-category-Precategory C) + where + + is-unfixed-replete-subprecategory-is-category-Subprecategory : + {x : obj-Subprecategory C P} + {y : obj-Precategory C} + (f : iso-Precategory C (inclusion-obj-Subprecategory C P x) y) → + is-in-iso-Subprecategory C P f + is-unfixed-replete-subprecategory-is-category-Subprecategory {x} = + ind-iso-Category + ( C , is-category-C) + ( λ B e → is-in-iso-Subprecategory C P e) + ( is-in-iso-id-Subprecategory C P x) + + is-replete-subprecategory-is-category-Subprecategory : + is-replete-Subprecategory C P + is-replete-subprecategory-is-category-Subprecategory x y = + ind-iso-Category + ( C , is-category-C) + ( λ z e → + Σ ( is-in-obj-Subprecategory C P z) + ( λ z₀ → + is-in-iso-obj-subprecategory-Subprecategory C P {x} {z , z₀} e)) + ( pr2 (is-in-iso-id-Subprecategory C P x)) +``` + +### If a full subprecategory is closed under isomorphic objects then it is replete + +This remains to be formalized. + +### The inclusion functor of a replete subprecategory is pseudomonic + +This remains to be formalized. + +## See also + +- Every [subcategory](category-theory.subcategories.md) is replete. + +- Because of universe polymorphism, + [large subcategories](category-theory.large-subcategories.md) are not large + replete by construction, although they are levelwise replete. + +## External links + +- [replete subcategory](https://ncatlab.org/nlab/show/replete+replete-subprecategory) + at nlab +- [Isomorphism-closed subcategory](https://en.wikipedia.org/wiki/Isomorphism-closed_subcategory) + at Wikipedia +- [isomorphism-closed subcategory](https://www.wikidata.org/wiki/Q6086096) at + Wikidata diff --git a/src/category-theory/representable-functors-categories.lagda.md b/src/category-theory/representable-functors-categories.lagda.md index 289bb57e8b..22014d5f9d 100644 --- a/src/category-theory/representable-functors-categories.lagda.md +++ b/src/category-theory/representable-functors-categories.lagda.md @@ -38,8 +38,8 @@ associativity and the left unit law for the category `C`. representable-functor-Category : {l1 l2 : Level} (C : Category l1 l2) (c : obj-Category C) → functor-Category C (Set-Category l2) -representable-functor-Category C c = - representable-functor-Precategory (precategory-Category C) c +representable-functor-Category C = + representable-functor-Precategory (precategory-Category C) ``` ## Natural transformations between representable functors @@ -51,7 +51,7 @@ components `hom c x → hom b x` are defined by precomposition with `f`. ```agda representable-natural-transformation-Category : - {l1 l2 : Level} (C : Category l1 l2) (b c : obj-Category C) + {l1 l2 : Level} (C : Category l1 l2) {b c : obj-Category C} (f : hom-Category C b c) → natural-transformation-Category ( C) diff --git a/src/category-theory/representable-functors-precategories.lagda.md b/src/category-theory/representable-functors-precategories.lagda.md index 5015e85c0f..4d7cf174de 100644 --- a/src/category-theory/representable-functors-precategories.lagda.md +++ b/src/category-theory/representable-functors-precategories.lagda.md @@ -7,14 +7,18 @@ module category-theory.representable-functors-precategories where
Imports ```agda +open import category-theory.copresheaf-categories open import category-theory.functors-precategories +open import category-theory.maps-precategories open import category-theory.natural-transformations-functors-precategories +open import category-theory.opposite-precategories open import category-theory.precategories open import foundation.category-of-sets open import foundation.dependent-pair-types open import foundation.function-extensionality open import foundation.homotopies +open import foundation.sets open import foundation.universe-levels ``` @@ -28,8 +32,8 @@ there is a [functor](category-theory.functors-precategories.md) from `C` to the that: - sends an object `x` of `C` to the [set](foundation-core.sets.md) `hom c x` and -- sends a morphism `g : hom x y` of `C` to the function `hom c x → hom c y` - defined by postcomposition with `g`. +- sends a morphism `f : hom x y` of `C` to the function `hom c x → hom c y` + defined by postcomposition with `f`. The functoriality axioms follow, by [function extensionality](foundation.function-extensionality.md), from @@ -38,16 +42,47 @@ associativity and the left unit law for the precategory `C`. ## Definition ```agda -representable-functor-Precategory : - {l1 l2 : Level} (C : Precategory l1 l2) (c : obj-Precategory C) → - functor-Precategory C (Set-Precategory l2) -pr1 (representable-functor-Precategory C c) = hom-set-Precategory C c -pr1 (pr2 (representable-functor-Precategory C c)) g = - postcomp-hom-Precategory C g c -pr1 (pr2 (pr2 (representable-functor-Precategory C c))) h g = - eq-htpy (associative-comp-hom-Precategory C h g) -pr2 (pr2 (pr2 (representable-functor-Precategory C c))) _ = - eq-htpy (left-unit-law-comp-hom-Precategory C) +module _ + {l1 l2 : Level} (C : Precategory l1 l2) (c : obj-Precategory C) + where + + obj-representable-functor-Precategory : obj-Precategory C → Set l2 + obj-representable-functor-Precategory = hom-set-Precategory C c + + hom-representable-functor-Precategory : + {x y : obj-Precategory C} (f : hom-Precategory C x y) → + hom-Precategory C c x → hom-Precategory C c y + hom-representable-functor-Precategory f = postcomp-hom-Precategory C f c + + representable-map-Precategory : map-Precategory C (Set-Precategory l2) + pr1 representable-map-Precategory = obj-representable-functor-Precategory + pr2 representable-map-Precategory = hom-representable-functor-Precategory + + preserves-comp-representable-functor-Precategory : + preserves-comp-hom-map-Precategory + ( C) + ( Set-Precategory l2) + ( representable-map-Precategory) + preserves-comp-representable-functor-Precategory g f = + eq-htpy (associative-comp-hom-Precategory C g f) + + preserves-id-representable-functor-Precategory : + preserves-id-hom-map-Precategory + ( C) + ( Set-Precategory l2) + ( representable-map-Precategory) + preserves-id-representable-functor-Precategory x = + eq-htpy (left-unit-law-comp-hom-Precategory C) + + representable-functor-Precategory : functor-Precategory C (Set-Precategory l2) + pr1 representable-functor-Precategory = + obj-representable-functor-Precategory + pr1 (pr2 representable-functor-Precategory) = + hom-representable-functor-Precategory + pr1 (pr2 (pr2 representable-functor-Precategory)) = + preserves-comp-representable-functor-Precategory + pr2 (pr2 (pr2 representable-functor-Precategory)) = + preserves-id-representable-functor-Precategory ``` ## Natural transformations between representable functors @@ -58,16 +93,59 @@ from the functor represented by `c` to the functor represented by `b`. Its components `hom c x → hom b x` are defined by precomposition with `f`. ```agda -representable-natural-transformation-Precategory : - {l1 l2 : Level} (C : Precategory l1 l2) (b c : obj-Precategory C) - (f : hom-Precategory C b c) → - natural-transformation-Precategory - ( C) - ( Set-Precategory l2) - ( representable-functor-Precategory C c) - ( representable-functor-Precategory C b) -pr1 (representable-natural-transformation-Precategory C b c f) = - precomp-hom-Precategory C f -pr2 (representable-natural-transformation-Precategory C b c f) h = - eq-htpy (inv-htpy (λ g → associative-comp-hom-Precategory C h g f)) +module _ + {l1 l2 : Level} (C : Precategory l1 l2) + {b c : obj-Precategory C} (f : hom-Precategory C b c) + where + + hom-family-representable-natural-transformation-Precategory : + hom-family-functor-Precategory + ( C) + ( Set-Precategory l2) + ( representable-functor-Precategory C c) + ( representable-functor-Precategory C b) + hom-family-representable-natural-transformation-Precategory = + precomp-hom-Precategory C f + + is-natural-transformation-representable-natural-transformation-Precategory : + is-natural-transformation-Precategory + ( C) + ( Set-Precategory l2) + ( representable-functor-Precategory C c) + ( representable-functor-Precategory C b) + ( hom-family-representable-natural-transformation-Precategory) + is-natural-transformation-representable-natural-transformation-Precategory h = + eq-htpy (inv-htpy (λ g → associative-comp-hom-Precategory C h g f)) + + representable-natural-transformation-Precategory : + natural-transformation-Precategory + ( C) + ( Set-Precategory l2) + ( representable-functor-Precategory C c) + ( representable-functor-Precategory C b) + pr1 (representable-natural-transformation-Precategory) = + hom-family-representable-natural-transformation-Precategory + pr2 (representable-natural-transformation-Precategory) = + is-natural-transformation-representable-natural-transformation-Precategory +``` + +## Properties + +### Taking representable functors defines a functor into the presheaf category + +```agda +module _ + {l1 l2 : Level} (C : Precategory l1 l2) + where + + map-representable-functor-copresheaf-Precategory : + map-Precategory + ( opposite-Precategory C) + ( copresheaf-Precategory C l2) + pr1 map-representable-functor-copresheaf-Precategory = + representable-functor-Precategory C + pr2 map-representable-functor-copresheaf-Precategory = + representable-natural-transformation-Precategory C ``` + +It remains to show that this map is functorial. diff --git a/src/category-theory/rigid-objects-categories.lagda.md b/src/category-theory/rigid-objects-categories.lagda.md new file mode 100644 index 0000000000..6b27878c52 --- /dev/null +++ b/src/category-theory/rigid-objects-categories.lagda.md @@ -0,0 +1,81 @@ +# Rigid objects in a category + +```agda +module category-theory.rigid-objects-categories where +``` + +
Imports + +```agda +open import category-theory.categories +open import category-theory.isomorphisms-in-categories +open import category-theory.rigid-objects-precategories + +open import foundation.contractible-types +open import foundation.dependent-pair-types +open import foundation.function-types +open import foundation.identity-types +open import foundation.propositions +open import foundation.universe-levels + +open import group-theory.trivial-groups +``` + +
+ +## Idea + +A **rigid object** in a [category](category-theory.categories.md) is an object +whose [automorphism group](group-theory.automorphism-groups.md) is +[trivial](group-theory.trivial-groups.md). + +## Definitions + +### The predicate of being rigid + +```agda +module _ + {l1 l2 : Level} (C : Category l1 l2) (x : obj-Category C) + where + + is-rigid-obj-prop-Category : Prop l2 + is-rigid-obj-prop-Category = + is-rigid-obj-prop-Precategory (precategory-Category C) x + + is-rigid-obj-Category : UU l2 + is-rigid-obj-Category = type-Prop is-rigid-obj-prop-Category + + is-prop-is-rigid-obj-Category : is-prop is-rigid-obj-Category + is-prop-is-rigid-obj-Category = + is-prop-type-Prop is-rigid-obj-prop-Category +``` + +### The type of rigid objects in a category + +```agda +rigid-obj-Category : {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2) +rigid-obj-Category C = + rigid-obj-Precategory (precategory-Category C) + +module _ + {l1 l2 : Level} (C : Category l1 l2) + where + + obj-rigid-obj-Category : rigid-obj-Category C → obj-Category C + obj-rigid-obj-Category = obj-rigid-obj-Precategory (precategory-Category C) + + is-rigid-rigid-obj-Category : + (x : rigid-obj-Category C) → + is-rigid-obj-Category C (obj-rigid-obj-Category x) + is-rigid-rigid-obj-Category = + is-rigid-rigid-obj-Precategory (precategory-Category C) +``` + +## See also + +- Every object in a category is rigid if and only if it is + [gaunt](category-theory.gaunt-categories.md). + +## External links + +- [rigid object](https://ncatlab.org/nlab/show/rigid+object) at nlab diff --git a/src/category-theory/rigid-objects-precategories.lagda.md b/src/category-theory/rigid-objects-precategories.lagda.md new file mode 100644 index 0000000000..6d7a1909c5 --- /dev/null +++ b/src/category-theory/rigid-objects-precategories.lagda.md @@ -0,0 +1,72 @@ +# Rigid objects in a precategory + +```agda +module category-theory.rigid-objects-precategories where +``` + +
Imports + +```agda +open import category-theory.isomorphisms-in-precategories +open import category-theory.precategories + +open import foundation.contractible-types +open import foundation.dependent-pair-types +open import foundation.function-types +open import foundation.identity-types +open import foundation.propositions +open import foundation.universe-levels + +open import group-theory.trivial-groups +``` + +
+ +## Idea + +A **rigid object** in a [precategory](category-theory.precategories.md) is an +object whose [automorphism group](group-theory.automorphism-groups.md) is +[trivial](group-theory.trivial-groups.md). + +## Definitions + +### The predicate of being rigid + +```agda +module _ + {l1 l2 : Level} (C : Precategory l1 l2) (x : obj-Precategory C) + where + + is-rigid-obj-prop-Precategory : Prop l2 + is-rigid-obj-prop-Precategory = is-contr-Prop (iso-Precategory C x x) + + is-rigid-obj-Precategory : UU l2 + is-rigid-obj-Precategory = type-Prop is-rigid-obj-prop-Precategory + + is-prop-is-rigid-obj-Precategory : is-prop is-rigid-obj-Precategory + is-prop-is-rigid-obj-Precategory = + is-prop-type-Prop is-rigid-obj-prop-Precategory +``` + +### The type of rigid objects in a precategory + +```agda +rigid-obj-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2) +rigid-obj-Precategory C = Σ (obj-Precategory C) (is-rigid-obj-Precategory C) + +module _ + {l1 l2 : Level} (C : Precategory l1 l2) + where + + obj-rigid-obj-Precategory : rigid-obj-Precategory C → obj-Precategory C + obj-rigid-obj-Precategory = pr1 + + is-rigid-rigid-obj-Precategory : + (x : rigid-obj-Precategory C) → + is-rigid-obj-Precategory C (obj-rigid-obj-Precategory x) + is-rigid-rigid-obj-Precategory = pr2 +``` + +## External links + +- [rigid object](https://ncatlab.org/nlab/show/rigid+object) at nlab diff --git a/src/category-theory/strict-categories.lagda.md b/src/category-theory/strict-categories.lagda.md index 90d251a46f..91492df8c8 100644 --- a/src/category-theory/strict-categories.lagda.md +++ b/src/category-theory/strict-categories.lagda.md @@ -7,15 +7,15 @@ module category-theory.strict-categories where
Imports ```agda +open import category-theory.categories open import category-theory.composition-operations-on-binary-families-of-sets open import category-theory.isomorphisms-in-precategories open import category-theory.nonunital-precategories open import category-theory.precategories open import category-theory.preunivalent-categories -open import foundation.1-types +open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.equivalences open import foundation.identity-types open import foundation.injective-maps open import foundation.propositions @@ -30,9 +30,10 @@ open import foundation.universe-levels A **strict category** is a [precategory](category-theory.precategories.md) for which the type of objects form a [set](foundation-core.sets.md). Such categories -are the set-theoretic analogue to (univalent) -[categories](category-theory.categories.md), and have the defect that strict -categorical constructions may generally fail to be invariant under isomorphisms. +are the set-theoretic analogue to +[(univalent) categories](category-theory.categories.md), and have the defect +that strict categorical constructions may generally fail to be invariant under +equivalences. ## Definitions @@ -48,7 +49,45 @@ module _ is-set-Prop (obj-Precategory C) is-strict-category-Precategory : UU l1 - is-strict-category-Precategory = type-Prop is-strict-category-prop-Precategory + is-strict-category-Precategory = + type-Prop is-strict-category-prop-Precategory +``` + +### The predicate on preunivalent categories of being a strict category + +```agda +module _ + {l1 l2 : Level} (C : Preunivalent-Category l1 l2) + where + + is-strict-category-prop-Preunivalent-Category : Prop l1 + is-strict-category-prop-Preunivalent-Category = + is-strict-category-prop-Precategory (precategory-Preunivalent-Category C) + + is-strict-category-Preunivalent-Category : UU l1 + is-strict-category-Preunivalent-Category = + type-Prop is-strict-category-prop-Preunivalent-Category +``` + +### The predicate on categories of being a strict category + +We note that [(univalent) categories](category-theory.categories.md) that are +strict form a very restricted class of strict categories where every +[isomorphism](category-theory.isomorphisms-in-categories.md)-set is a +[proposition](foundation-core.propositions.md). Such a category is called +[gaunt](category-theory.gaunt-categories.md). + +```agda +module _ + {l1 l2 : Level} (C : Category l1 l2) + where + + is-strict-category-prop-Category : Prop l1 + is-strict-category-prop-Category = + is-strict-category-prop-Precategory (precategory-Category C) + + is-strict-category-Category : UU l1 + is-strict-category-Category = type-Prop is-strict-category-prop-Category ``` ### The type of strict categories @@ -162,6 +201,36 @@ module _ pr2 preunivalent-category-Strict-Category = is-preunivalent-Strict-Category ``` +### The total hom-set of a strict category + +```agda +module _ + {l1 l2 : Level} (C : Strict-Category l1 l2) + where + + total-hom-Strict-Category : UU (l1 ⊔ l2) + total-hom-Strict-Category = + total-hom-Precategory (precategory-Strict-Category C) + + obj-total-hom-Strict-Category : + total-hom-Strict-Category → obj-Strict-Category C × obj-Strict-Category C + obj-total-hom-Strict-Category = + obj-total-hom-Precategory (precategory-Strict-Category C) + + is-set-total-hom-Strict-Category : + is-set total-hom-Strict-Category + is-set-total-hom-Strict-Category = + is-trunc-total-hom-is-trunc-obj-Precategory + ( precategory-Strict-Category C) + ( is-set-obj-Strict-Category C) + + total-hom-set-Strict-Category : Set (l1 ⊔ l2) + total-hom-set-Strict-Category = + total-hom-truncated-type-is-trunc-obj-Precategory + ( precategory-Strict-Category C) + ( is-set-obj-Strict-Category C) +``` + ### Equalities induce morphisms ```agda @@ -201,3 +270,13 @@ postcomp-hom-Strict-Category C = - [Preunivalent categories](category-theory.preunivalent-categories.md) for the common generalization of (univalent) categories and strict categories. +- [Gaunt categories](category-theory.gaunt-categories.md) for the common + intersection of (univalent) categories and strict categories. + +## External links + +- [Strict Precategories](https://1lab.dev/Cat.Strict.html#strict-precategories) + at 1lab +- [strict category](https://ncatlab.org/nlab/show/strict+category) at nlab +- [Category (mathematics)]() + at Wikipedia diff --git a/src/category-theory/subcategories.lagda.md b/src/category-theory/subcategories.lagda.md index f359b8bf77..79218bd947 100644 --- a/src/category-theory/subcategories.lagda.md +++ b/src/category-theory/subcategories.lagda.md @@ -11,15 +11,25 @@ open import category-theory.categories open import category-theory.composition-operations-on-binary-families-of-sets open import category-theory.faithful-functors-precategories open import category-theory.functors-precategories +open import category-theory.isomorphisms-in-categories +open import category-theory.isomorphisms-in-precategories +open import category-theory.isomorphisms-in-subprecategories open import category-theory.maps-precategories open import category-theory.precategories +open import category-theory.replete-subprecategories open import category-theory.subprecategories +open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.embeddings +open import foundation.equivalences +open import foundation.functoriality-dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.homotopies open import foundation.identity-types open import foundation.propositions open import foundation.sets +open import foundation.subtype-identity-principle open import foundation.subtypes open import foundation.universe-levels ``` @@ -48,7 +58,7 @@ module _ {l1 l2 l3 l4 : Level} (C : Category l1 l2) (P₀ : subtype l3 (obj-Category C)) - (P₁ : (x y : obj-Category C) → subtype l4 (hom-Category C x y)) + (P₁ : subtype-hom-Precategory l4 (precategory-Category C) P₀) where contains-id-subtype-Category : UU (l1 ⊔ l3 ⊔ l4) @@ -88,12 +98,11 @@ module _ {l1 l2 l3 l4 : Level} (C : Category l1 l2) (P₀ : subtype l3 (obj-Category C)) - (P₁ : (x y : obj-Category C) → subtype l4 (hom-Category C x y)) + (P₁ : subtype-hom-Precategory l4 (precategory-Category C) P₀) where is-subcategory-Prop : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4) - is-subcategory-Prop = - is-subprecategory-Prop (precategory-Category C) P₀ P₁ + is-subcategory-Prop = is-subprecategory-Prop (precategory-Category C) P₀ P₁ is-subcategory : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) is-subcategory = type-Prop is-subcategory-Prop @@ -102,12 +111,16 @@ module _ is-prop-is-subcategory = is-prop-type-Prop is-subcategory-Prop contains-id-is-subcategory : - is-subcategory → contains-id-subtype-Category C P₀ P₁ - contains-id-is-subcategory = pr1 + is-subcategory → + contains-id-subtype-Category C P₀ P₁ + contains-id-is-subcategory = + contains-id-is-subprecategory (precategory-Category C) P₀ P₁ is-closed-under-composition-is-subcategory : - is-subcategory → is-closed-under-composition-subtype-Category C P₀ P₁ - is-closed-under-composition-is-subcategory = pr2 + is-subcategory → + is-closed-under-composition-subtype-Category C P₀ P₁ + is-closed-under-composition-is-subcategory = + is-closed-under-composition-is-subprecategory (precategory-Category C) P₀ P₁ ``` ### Subcategories @@ -118,7 +131,11 @@ Subcategory : (C : Category l1 l2) → UU (l1 ⊔ l2 ⊔ lsuc l3 ⊔ lsuc l4) Subcategory l3 l4 C = Subprecategory l3 l4 (precategory-Category C) +``` + +#### Objects in subcategories +```agda module _ {l1 l2 l3 l4 : Level} (C : Category l1 l2) @@ -126,7 +143,8 @@ module _ where subtype-obj-Subcategory : subtype l3 (obj-Category C) - subtype-obj-Subcategory = pr1 P + subtype-obj-Subcategory = + subtype-obj-Subprecategory (precategory-Category C) P obj-Subcategory : UU (l1 ⊔ l3) obj-Subcategory = type-subtype subtype-obj-Subcategory @@ -139,63 +157,133 @@ module _ is-prop-is-in-obj-Subcategory : (x : obj-Category C) → is-prop (is-in-obj-Subcategory x) - is-prop-is-in-obj-Subcategory = is-prop-is-in-subtype subtype-obj-Subcategory + is-prop-is-in-obj-Subcategory = + is-prop-is-in-subtype subtype-obj-Subcategory is-in-obj-inclusion-obj-Subcategory : (x : obj-Subcategory) → is-in-obj-Subcategory (inclusion-obj-Subcategory x) is-in-obj-inclusion-obj-Subcategory = is-in-subtype-inclusion-subtype subtype-obj-Subcategory +``` - subtype-hom-Subcategory : - (x y : obj-Category C) → subtype l4 (hom-Category C x y) - subtype-hom-Subcategory = pr1 (pr2 P) +#### Morphisms in subcategories - hom-Subcategory : (x y : obj-Subcategory) → UU (l2 ⊔ l4) - hom-Subcategory = hom-Subprecategory (precategory-Category C) P +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Category l1 l2) + (P : Subcategory l3 l4 C) + where + + subtype-hom-Subcategory : + subtype-hom-Precategory l4 + ( precategory-Category C) + ( subtype-obj-Subcategory C P) + subtype-hom-Subcategory = + subtype-hom-Subprecategory (precategory-Category C) P + + subtype-hom-obj-subcategory-Subcategory : + (x y : obj-Subcategory C P) → + subtype l4 + ( hom-Category C + ( inclusion-obj-Subcategory C P x) + ( inclusion-obj-Subcategory C P y)) + subtype-hom-obj-subcategory-Subcategory = + subtype-hom-obj-subprecategory-Subprecategory (precategory-Category C) P + + hom-Subcategory : (x y : obj-Subcategory C P) → UU (l2 ⊔ l4) + hom-Subcategory x y = + type-subtype (subtype-hom-obj-subcategory-Subcategory x y) inclusion-hom-Subcategory : - (x y : obj-Subcategory) → + (x y : obj-Subcategory C P) → hom-Subcategory x y → hom-Category C - ( inclusion-obj-Subcategory x) - ( inclusion-obj-Subcategory y) - inclusion-hom-Subcategory = - inclusion-hom-Subprecategory (precategory-Category C) P + ( inclusion-obj-Subcategory C P x) + ( inclusion-obj-Subcategory C P y) + inclusion-hom-Subcategory x y = + inclusion-subtype (subtype-hom-obj-subcategory-Subcategory x y) +``` +The predicate on a morphism between subobjects of being contained in the +subcategory: + +```agda + is-in-hom-obj-subcategory-Subcategory : + ( x y : obj-Subcategory C P) → + hom-Category C + ( inclusion-obj-Subcategory C P x) + ( inclusion-obj-Subcategory C P y) → + UU l4 + is-in-hom-obj-subcategory-Subcategory x y = + is-in-subtype (subtype-hom-obj-subcategory-Subcategory x y) + + is-prop-is-in-hom-obj-subcategory-Subcategory : + ( x y : obj-Subcategory C P) + ( f : + hom-Category C + ( inclusion-obj-Subcategory C P x) + ( inclusion-obj-Subcategory C P y)) → + is-prop (is-in-hom-obj-subcategory-Subcategory x y f) + is-prop-is-in-hom-obj-subcategory-Subcategory x y = + is-prop-is-in-subtype (subtype-hom-obj-subcategory-Subcategory x y) +``` + +The predicate on a morphism between any objects of being contained in the +subcategory: + +```agda is-in-hom-Subcategory : - (x y : obj-Category C) (f : hom-Category C x y) → UU l4 - is-in-hom-Subcategory x y = is-in-subtype (subtype-hom-Subcategory x y) + (x y : obj-Category C) (f : hom-Category C x y) → UU (l3 ⊔ l4) + is-in-hom-Subcategory = + is-in-hom-Subprecategory (precategory-Category C) P is-prop-is-in-hom-Subcategory : (x y : obj-Category C) (f : hom-Category C x y) → is-prop (is-in-hom-Subcategory x y f) - is-prop-is-in-hom-Subcategory x y = - is-prop-is-in-subtype (subtype-hom-Subcategory x y) + is-prop-is-in-hom-Subcategory = + is-prop-is-in-hom-Subprecategory (precategory-Category C) P + + is-in-hom-obj-subcategory-inclusion-hom-Subcategory : + (x y : obj-Subcategory C P) + (f : hom-Subcategory x y) → + is-in-hom-obj-subcategory-Subcategory x y + ( inclusion-hom-Subcategory x y f) + is-in-hom-obj-subcategory-inclusion-hom-Subcategory = + is-in-hom-obj-subprecategory-inclusion-hom-Subprecategory + ( precategory-Category C) P is-in-hom-inclusion-hom-Subcategory : - (x y : obj-Subcategory) (f : hom-Subcategory x y) → + (x y : obj-Subcategory C P) + (f : hom-Subcategory x y) → is-in-hom-Subcategory - ( inclusion-obj-Subcategory x) - ( inclusion-obj-Subcategory y) + ( inclusion-obj-Subcategory C P x) + ( inclusion-obj-Subcategory C P y) ( inclusion-hom-Subcategory x y f) is-in-hom-inclusion-hom-Subcategory = is-in-hom-inclusion-hom-Subprecategory (precategory-Category C) P +``` + +#### Subcategories are subcategories +```agda is-subcategory-Subcategory : - is-subcategory C subtype-obj-Subcategory subtype-hom-Subcategory - is-subcategory-Subcategory = pr2 (pr2 P) + is-subcategory C + ( subtype-obj-Subcategory C P) (subtype-hom-Subcategory) + is-subcategory-Subcategory = + is-subprecategory-Subprecategory (precategory-Category C) P contains-id-Subcategory : contains-id-subtype-Category C - ( subtype-obj-Subcategory) + ( subtype-obj-Subcategory C P) ( subtype-hom-Subcategory) contains-id-Subcategory = contains-id-Subprecategory (precategory-Category C) P is-closed-under-composition-Subcategory : is-closed-under-composition-subtype-Category C - ( subtype-obj-Subcategory) + ( subtype-obj-Subcategory C P) ( subtype-hom-Subcategory) is-closed-under-composition-Subcategory = is-closed-under-composition-Subprecategory (precategory-Category C) P @@ -211,23 +299,25 @@ module _ where hom-set-Subcategory : (x y : obj-Subcategory C P) → Set (l2 ⊔ l4) - hom-set-Subcategory = hom-set-Subprecategory (precategory-Category C) P + hom-set-Subcategory = + hom-set-Subprecategory (precategory-Category C) P is-set-hom-Subcategory : (x y : obj-Subcategory C P) → is-set (hom-Subcategory C P x y) - is-set-hom-Subcategory = is-set-hom-Subprecategory (precategory-Category C) P + is-set-hom-Subcategory x y = is-set-type-Set (hom-set-Subcategory x y) id-hom-Subcategory : {x : obj-Subcategory C P} → hom-Subcategory C P x x - id-hom-Subcategory {x} = id-hom-Subprecategory (precategory-Category C) P {x} + id-hom-Subcategory = + id-hom-Subprecategory (precategory-Category C) P comp-hom-Subcategory : {x y z : obj-Subcategory C P} → hom-Subcategory C P y z → hom-Subcategory C P x y → hom-Subcategory C P x z - comp-hom-Subcategory {x} {y} {z} = - comp-hom-Subprecategory (precategory-Category C) P {x} {y} {z} + comp-hom-Subcategory = + comp-hom-Subprecategory (precategory-Category C) P associative-comp-hom-Subcategory : {x y z w : obj-Subcategory C P} @@ -272,10 +362,6 @@ module _ precategory-Subprecategory (precategory-Category C) P ``` -### The underlying category of a subcategory - -It remains to show that subprecategories of categories are categories. - ### The inclusion functor of a subcategory ```agda @@ -303,7 +389,9 @@ module _ inclusion-Subcategory = inclusion-Subprecategory (precategory-Category C) P ``` -### Isomorphisms in subcategories +## Properties + +### Subcategories are replete ```agda module _ @@ -312,19 +400,63 @@ module _ (P : Subcategory l3 l4 C) where - is-iso-Subcategory : - {x y : obj-Subcategory C P} → hom-Subcategory C P x y → UU (l2 ⊔ l4) - is-iso-Subcategory {x} {y} = - is-iso-Subprecategory (precategory-Category C) P {x} {y} - - iso-Subcategory : - (x y : obj-Subcategory C P) → UU (l2 ⊔ l4) - iso-Subcategory = iso-Subprecategory (precategory-Category C) P + is-replete-Subcategory : is-replete-Subprecategory (precategory-Category C) P + is-replete-Subcategory = + is-replete-subprecategory-is-category-Subprecategory + ( precategory-Category C) + ( P) + ( is-category-Category C) + + compute-iso-Subcategory : + {x y : obj-Subcategory C P} → + iso-Category C + ( inclusion-obj-Subcategory C P x) + ( inclusion-obj-Subcategory C P y) ≃ + iso-Subprecategory (precategory-Category C) P x y + compute-iso-Subcategory {x} {y} = + compute-iso-is-replete-Subprecategory + ( precategory-Category C) P is-replete-Subcategory x y + + inv-compute-iso-Subcategory : + {x y : obj-Subcategory C P} → + iso-Subprecategory (precategory-Category C) P x y ≃ + iso-Category C + ( inclusion-obj-Subcategory C P x) + ( inclusion-obj-Subcategory C P y) + inv-compute-iso-Subcategory {x} {y} = + inv-compute-iso-is-replete-Subprecategory + ( precategory-Category C) P is-replete-Subcategory x y ``` -## Properties +### Subcategories are categories -### The inclusion functor is faithful and an embedding on objects +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Category l1 l2) + (P : Subcategory l3 l4 C) + where + + is-category-Subcategory : + is-category-Precategory (precategory-Subcategory C P) + is-category-Subcategory x = + fundamental-theorem-id + ( is-contr-equiv _ + ( equiv-tot (λ y → inv-compute-iso-Subcategory C P {x} {y})) + ( is-torsorial-Eq-subtype + ( is-torsorial-iso-Category C (inclusion-obj-Subcategory C P x)) + ( is-prop-is-in-obj-Subcategory C P) + ( inclusion-obj-Subcategory C P x) + ( id-iso-Category C) + ( is-in-obj-inclusion-obj-Subcategory C P x))) + ( iso-eq-Precategory (precategory-Subcategory C P) x) + + category-Subcategory : Category (l1 ⊔ l3) (l2 ⊔ l4) + pr1 category-Subcategory = precategory-Subcategory C P + pr2 category-Subcategory = is-category-Subcategory +``` + +### The inclusion functor is an embedding on objects and hom-sets ```agda module _ @@ -333,20 +465,24 @@ module _ (P : Subcategory l3 l4 C) where - is-faithful-inclusion-Category : + is-faithful-inclusion-Subcategory : is-faithful-functor-Precategory ( precategory-Subcategory C P) ( precategory-Category C) ( inclusion-Subcategory C P) - is-faithful-inclusion-Category = + is-faithful-inclusion-Subcategory = is-faithful-inclusion-Subprecategory (precategory-Category C) P - is-emb-obj-inclusion-Category : + is-emb-obj-inclusion-Subcategory : is-emb ( obj-functor-Precategory ( precategory-Subcategory C P) ( precategory-Category C) ( inclusion-Subcategory C P)) - is-emb-obj-inclusion-Category = + is-emb-obj-inclusion-Subcategory = is-emb-obj-inclusion-Subprecategory (precategory-Category C) P ``` + +### The inclusion functor is pseudomonic + +This is another consequence of repleteness. diff --git a/src/category-theory/subprecategories.lagda.md b/src/category-theory/subprecategories.lagda.md index c5a972b724..fa98a332c9 100644 --- a/src/category-theory/subprecategories.lagda.md +++ b/src/category-theory/subprecategories.lagda.md @@ -7,22 +7,18 @@ module category-theory.subprecategories where
Imports ```agda -open import category-theory.categories open import category-theory.composition-operations-on-binary-families-of-sets open import category-theory.faithful-functors-precategories open import category-theory.functors-precategories -open import category-theory.isomorphism-induction-categories -open import category-theory.isomorphisms-in-precategories open import category-theory.maps-precategories open import category-theory.precategories open import foundation.dependent-pair-types open import foundation.embeddings -open import foundation.function-types open import foundation.identity-types +open import foundation.iterated-dependent-product-types open import foundation.propositions open import foundation.sets -open import foundation.subsingleton-induction open import foundation.subtypes open import foundation.universe-levels ``` @@ -44,24 +40,39 @@ objects in `P₀` and is closed under composition. ## Definition +### Sub-hom-families + +```agda +subtype-hom-Precategory : + {l1 l2 l3 : Level} (l4 : Level) + (C : Precategory l1 l2) + (P₀ : subtype l3 (obj-Precategory C)) → UU (l1 ⊔ l2 ⊔ l3 ⊔ lsuc l4) +subtype-hom-Precategory l4 C P₀ = + (x y : obj-Precategory C) → is-in-subtype P₀ x → is-in-subtype P₀ y → + subtype l4 (hom-Precategory C x y) +``` + +### Categorical predicates on sub-hom-families + ```agda module _ {l1 l2 l3 l4 : Level} (C : Precategory l1 l2) (P₀ : subtype l3 (obj-Precategory C)) - (P₁ : (x y : obj-Precategory C) → subtype l4 (hom-Precategory C x y)) + (P₁ : subtype-hom-Precategory l4 C P₀) where contains-id-subtype-Precategory : UU (l1 ⊔ l3 ⊔ l4) contains-id-subtype-Precategory = (x : obj-Precategory C) → - is-in-subtype P₀ x → is-in-subtype (P₁ x x) (id-hom-Precategory C) + (x₀ : is-in-subtype P₀ x) → + is-in-subtype (P₁ x x x₀ x₀) (id-hom-Precategory C) is-prop-contains-id-subtype-Precategory : is-prop contains-id-subtype-Precategory is-prop-contains-id-subtype-Precategory = - is-prop-Π² - ( λ x _ → is-prop-is-in-subtype (P₁ x x) (id-hom-Precategory C)) + is-prop-iterated-Π 2 + ( λ x x₀ → is-prop-is-in-subtype (P₁ x x x₀ x₀) (id-hom-Precategory C)) contains-id-prop-subtype-Precategory : Prop (l1 ⊔ l3 ⊔ l4) pr1 contains-id-prop-subtype-Precategory = @@ -74,19 +85,19 @@ module _ (x y z : obj-Precategory C) → (g : hom-Precategory C y z) → (f : hom-Precategory C x y) → - is-in-subtype P₀ x → - is-in-subtype P₀ y → - is-in-subtype P₀ z → - is-in-subtype (P₁ y z) g → - is-in-subtype (P₁ x y) f → - is-in-subtype (P₁ x z) (comp-hom-Precategory C g f) + (x₀ : is-in-subtype P₀ x) → + (y₀ : is-in-subtype P₀ y) → + (z₀ : is-in-subtype P₀ z) → + is-in-subtype (P₁ y z y₀ z₀) g → + is-in-subtype (P₁ x y x₀ y₀) f → + is-in-subtype (P₁ x z x₀ z₀) (comp-hom-Precategory C g f) is-prop-is-closed-under-composition-subtype-Precategory : is-prop is-closed-under-composition-subtype-Precategory is-prop-is-closed-under-composition-subtype-Precategory = - is-prop-Π¹⁰ - ( λ x y z g f _ _ _ _ _ → - is-prop-is-in-subtype (P₁ x z) (comp-hom-Precategory C g f)) + is-prop-iterated-Π 10 + ( λ x y z g f x₀ _ z₀ _ _ → + is-prop-is-in-subtype (P₁ x z x₀ z₀) (comp-hom-Precategory C g f)) is-closed-under-composition-prop-subtype-Precategory : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4) @@ -103,7 +114,7 @@ module _ {l1 l2 l3 l4 : Level} (C : Precategory l1 l2) (P₀ : subtype l3 (obj-Precategory C)) - (P₁ : (x y : obj-Precategory C) → subtype l4 (hom-Precategory C x y)) + (P₁ : subtype-hom-Precategory l4 C P₀) where is-subprecategory-Prop : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4) @@ -137,9 +148,13 @@ Subprecategory : Subprecategory l3 l4 C = Σ ( subtype l3 (obj-Precategory C)) ( λ P₀ → - Σ ( (x y : obj-Precategory C) → subtype l4 (hom-Precategory C x y)) + Σ ( subtype-hom-Precategory l4 C P₀) ( is-subprecategory C P₀)) +``` +#### Objects in subprecategories + +```agda module _ {l1 l2 l3 l4 : Level} (C : Precategory l1 l2) @@ -168,73 +183,145 @@ module _ is-in-obj-Subprecategory (inclusion-obj-Subprecategory x) is-in-obj-inclusion-obj-Subprecategory = is-in-subtype-inclusion-subtype subtype-obj-Subprecategory +``` + +#### Morphisms in subprecategories + +```agda +module _ + {l1 l2 l3 l4 : Level} + (C : Precategory l1 l2) + (P : Subprecategory l3 l4 C) + where subtype-hom-Subprecategory : - (x y : obj-Precategory C) → subtype l4 (hom-Precategory C x y) + subtype-hom-Precategory l4 C (subtype-obj-Subprecategory C P) subtype-hom-Subprecategory = pr1 (pr2 P) - hom-Subprecategory : (x y : obj-Subprecategory) → UU (l2 ⊔ l4) + subtype-hom-obj-subprecategory-Subprecategory : + (x y : obj-Subprecategory C P) → + subtype l4 + ( hom-Precategory C + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y)) + subtype-hom-obj-subprecategory-Subprecategory x y = + subtype-hom-Subprecategory + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y) + ( is-in-obj-inclusion-obj-Subprecategory C P x) + ( is-in-obj-inclusion-obj-Subprecategory C P y) + + hom-Subprecategory : (x y : obj-Subprecategory C P) → UU (l2 ⊔ l4) hom-Subprecategory x y = - type-subtype - ( subtype-hom-Subprecategory - ( inclusion-obj-Subprecategory x) - ( inclusion-obj-Subprecategory y)) + type-subtype (subtype-hom-obj-subprecategory-Subprecategory x y) inclusion-hom-Subprecategory : - (x y : obj-Subprecategory) → + (x y : obj-Subprecategory C P) → hom-Subprecategory x y → hom-Precategory C - ( inclusion-obj-Subprecategory x) - ( inclusion-obj-Subprecategory y) + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y) inclusion-hom-Subprecategory x y = - inclusion-subtype - ( subtype-hom-Subprecategory - ( inclusion-obj-Subprecategory x) - ( inclusion-obj-Subprecategory y)) + inclusion-subtype (subtype-hom-obj-subprecategory-Subprecategory x y) +``` + +The predicate on a morphism between subobjects of being contained in the +subprecategory: + +```agda + is-in-hom-obj-subprecategory-Subprecategory : + ( x y : obj-Subprecategory C P) → + hom-Precategory C + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y) → + UU l4 + is-in-hom-obj-subprecategory-Subprecategory x y = + is-in-subtype (subtype-hom-obj-subprecategory-Subprecategory x y) + + is-prop-is-in-hom-obj-subprecategory-Subprecategory : + ( x y : obj-Subprecategory C P) + ( f : + hom-Precategory C + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y)) → + is-prop (is-in-hom-obj-subprecategory-Subprecategory x y f) + is-prop-is-in-hom-obj-subprecategory-Subprecategory x y = + is-prop-is-in-subtype (subtype-hom-obj-subprecategory-Subprecategory x y) +``` + +The predicate on a morphism between any objects of being contained in the +subprecategory: +```agda is-in-hom-Subprecategory : - (x y : obj-Precategory C) (f : hom-Precategory C x y) → UU l4 - is-in-hom-Subprecategory x y = is-in-subtype (subtype-hom-Subprecategory x y) + (x y : obj-Precategory C) (f : hom-Precategory C x y) → UU (l3 ⊔ l4) + is-in-hom-Subprecategory x y f = + Σ ( is-in-obj-Subprecategory C P x) + ( λ x₀ → + Σ ( is-in-obj-Subprecategory C P y) + ( λ y₀ → + is-in-hom-obj-subprecategory-Subprecategory (x , x₀) (y , y₀) f)) is-prop-is-in-hom-Subprecategory : (x y : obj-Precategory C) (f : hom-Precategory C x y) → is-prop (is-in-hom-Subprecategory x y f) - is-prop-is-in-hom-Subprecategory x y = - is-prop-is-in-subtype (subtype-hom-Subprecategory x y) + is-prop-is-in-hom-Subprecategory x y f = + is-prop-Σ + ( is-prop-is-in-obj-Subprecategory C P x) + ( λ x₀ → + is-prop-Σ + ( is-prop-is-in-obj-Subprecategory C P y) + ( λ y₀ → + is-prop-is-in-hom-obj-subprecategory-Subprecategory + ( x , x₀) (y , y₀) f)) + + is-in-hom-obj-subprecategory-inclusion-hom-Subprecategory : + (x y : obj-Subprecategory C P) + (f : hom-Subprecategory x y) → + is-in-hom-obj-subprecategory-Subprecategory x y + ( inclusion-hom-Subprecategory x y f) + is-in-hom-obj-subprecategory-inclusion-hom-Subprecategory x y f = pr2 f is-in-hom-inclusion-hom-Subprecategory : - (x y : obj-Subprecategory) (f : hom-Subprecategory x y) → + (x y : obj-Subprecategory C P) + (f : hom-Subprecategory x y) → is-in-hom-Subprecategory - ( inclusion-obj-Subprecategory x) - ( inclusion-obj-Subprecategory y) + ( inclusion-obj-Subprecategory C P x) + ( inclusion-obj-Subprecategory C P y) ( inclusion-hom-Subprecategory x y f) - is-in-hom-inclusion-hom-Subprecategory x y = - is-in-subtype-inclusion-subtype - ( subtype-hom-Subprecategory - ( inclusion-obj-Subprecategory x) - ( inclusion-obj-Subprecategory y)) + pr1 (is-in-hom-inclusion-hom-Subprecategory x y f) = + is-in-obj-inclusion-obj-Subprecategory C P x + pr1 (pr2 (is-in-hom-inclusion-hom-Subprecategory x y f)) = + is-in-obj-inclusion-obj-Subprecategory C P y + pr2 (pr2 (is-in-hom-inclusion-hom-Subprecategory x y f)) = + is-in-hom-obj-subprecategory-inclusion-hom-Subprecategory x y f +``` +#### Subprecategories are subprecategories + +```agda is-subprecategory-Subprecategory : - is-subprecategory C subtype-obj-Subprecategory subtype-hom-Subprecategory + is-subprecategory C + ( subtype-obj-Subprecategory C P) (subtype-hom-Subprecategory) is-subprecategory-Subprecategory = pr2 (pr2 P) contains-id-Subprecategory : contains-id-subtype-Precategory C - ( subtype-obj-Subprecategory) + ( subtype-obj-Subprecategory C P) ( subtype-hom-Subprecategory) contains-id-Subprecategory = contains-id-is-subprecategory C - ( subtype-obj-Subprecategory) + ( subtype-obj-Subprecategory C P) ( subtype-hom-Subprecategory) ( is-subprecategory-Subprecategory) is-closed-under-composition-Subprecategory : is-closed-under-composition-subtype-Precategory C - ( subtype-obj-Subprecategory) + ( subtype-obj-Subprecategory C P) ( subtype-hom-Subprecategory) is-closed-under-composition-Subprecategory = is-closed-under-composition-is-subprecategory C - ( subtype-obj-Subprecategory) + ( subtype-obj-Subprecategory C P) ( subtype-hom-Subprecategory) ( is-subprecategory-Subprecategory) ``` @@ -254,9 +341,7 @@ module _ ( hom-set-Precategory C ( inclusion-obj-Subprecategory C P x) ( inclusion-obj-Subprecategory C P y)) - ( subtype-hom-Subprecategory C P - ( inclusion-obj-Subprecategory C P x) - ( inclusion-obj-Subprecategory C P y)) + ( subtype-hom-obj-subprecategory-Subprecategory C P x y) is-set-hom-Subprecategory : (x y : obj-Subprecategory C P) → is-set (hom-Subprecategory C P x y) @@ -289,8 +374,8 @@ module _ ( is-in-obj-inclusion-obj-Subprecategory C P x) ( is-in-obj-inclusion-obj-Subprecategory C P y) ( is-in-obj-inclusion-obj-Subprecategory C P z) - ( is-in-hom-inclusion-hom-Subprecategory C P y z g) - ( is-in-hom-inclusion-hom-Subprecategory C P x y f) + ( pr2 g) + ( pr2 f) associative-comp-hom-Subprecategory : {x y z w : obj-Subprecategory C P} @@ -305,7 +390,9 @@ module _ eq-type-subtype ( subtype-hom-Subprecategory C P ( inclusion-obj-Subprecategory C P x) - ( inclusion-obj-Subprecategory C P w)) + ( inclusion-obj-Subprecategory C P w) + ( is-in-obj-inclusion-obj-Subprecategory C P x) + ( is-in-obj-inclusion-obj-Subprecategory C P w)) ( associative-comp-hom-Precategory C ( inclusion-hom-Subprecategory C P z w h) ( inclusion-hom-Subprecategory C P y z g) @@ -319,7 +406,9 @@ module _ eq-type-subtype ( subtype-hom-Subprecategory C P ( inclusion-obj-Subprecategory C P x) - ( inclusion-obj-Subprecategory C P y)) + ( inclusion-obj-Subprecategory C P y) + ( is-in-obj-inclusion-obj-Subprecategory C P x) + ( is-in-obj-inclusion-obj-Subprecategory C P y)) ( left-unit-law-comp-hom-Precategory C ( inclusion-hom-Subprecategory C P x y f)) @@ -331,7 +420,9 @@ module _ eq-type-subtype ( subtype-hom-Subprecategory C P ( inclusion-obj-Subprecategory C P x) - ( inclusion-obj-Subprecategory C P y)) + ( inclusion-obj-Subprecategory C P y) + ( is-in-obj-inclusion-obj-Subprecategory C P x) + ( is-in-obj-inclusion-obj-Subprecategory C P y)) ( right-unit-law-comp-hom-Precategory C ( inclusion-hom-Subprecategory C P x y f)) @@ -396,25 +487,6 @@ module _ is-functor-inclusion-Subprecategory ``` -### Isomorphisms in subprecategories - -```agda -module _ - {l1 l2 l3 l4 : Level} - (C : Precategory l1 l2) - (P : Subprecategory l3 l4 C) - where - - is-iso-Subprecategory : - {x y : obj-Subprecategory C P} → hom-Subprecategory C P x y → UU (l2 ⊔ l4) - is-iso-Subprecategory {x} {y} = - is-iso-Precategory (precategory-Subprecategory C P) {x} {y} - - iso-Subprecategory : - (x y : obj-Subprecategory C P) → UU (l2 ⊔ l4) - iso-Subprecategory = iso-Precategory (precategory-Subprecategory C P) -``` - ## Properties ### The inclusion functor is an embedding on objects and hom-sets @@ -435,7 +507,9 @@ module _ is-emb-inclusion-subtype ( subtype-hom-Subprecategory C P ( inclusion-obj-Subprecategory C P x) - ( inclusion-obj-Subprecategory C P y)) + ( inclusion-obj-Subprecategory C P y) + ( is-in-obj-inclusion-obj-Subprecategory C P x) + ( is-in-obj-inclusion-obj-Subprecategory C P y)) is-emb-obj-inclusion-Subprecategory : is-emb @@ -446,47 +520,3 @@ module _ is-emb-obj-inclusion-Subprecategory = is-emb-inclusion-subtype (subtype-obj-Subprecategory C P) ``` - -### Subprecategories of categories are categories - -```agda -module _ - {l1 l2 l3 l4 : Level} - (C : Precategory l1 l2) - (P : Subprecategory l3 l4 C) - (is-category-C : is-category-Precategory C) - {x y : obj-Subprecategory C P} - (f : hom-Subprecategory C P x y) - (is-iso-f : is-iso-Precategory C (inclusion-hom-Subprecategory C P x y f)) - where - - contains-is-iso-is-category-Subprecategory : is-iso-Subprecategory C P f - contains-is-iso-is-category-Subprecategory = - ind-iso-Category (C , is-category-C) - ( λ Y e → - ( p : is-in-obj-Subprecategory C P Y) - ( q : - is-in-hom-Subprecategory C P - ( inclusion-obj-Subprecategory C P x) - ( Y) - ( hom-iso-Precategory C e)) → - is-iso-Subprecategory C P {x} {Y , p} (hom-iso-Precategory C e , q)) - ( ( ind-subsingleton - ( is-prop-is-in-hom-Subprecategory C P - ( inclusion-obj-Subprecategory C P x) - ( inclusion-obj-Subprecategory C P x) - ( id-hom-Precategory C)) - ( contains-id-Subprecategory C P - ( inclusion-obj-Subprecategory C P x) - ( is-in-obj-inclusion-obj-Subprecategory C P x))) ∘ - ( ind-subsingleton - ( is-prop-is-in-obj-Subprecategory C P - ( inclusion-obj-Subprecategory C P x)) - ( is-in-obj-inclusion-obj-Subprecategory C P x) - ( is-iso-id-hom-Precategory (precategory-Subprecategory C P) {x}))) - ( inclusion-hom-Subprecategory C P x y f , is-iso-f) - ( is-in-obj-inclusion-obj-Subprecategory C P y) - ( is-in-hom-inclusion-hom-Subprecategory C P x y f) -``` - -It remains to show that subprecategories of categories indeed are categories. diff --git a/src/category-theory/terminal-objects-precategories.lagda.md b/src/category-theory/terminal-objects-precategories.lagda.md index 22d9b360c7..5a1fcb7d9c 100644 --- a/src/category-theory/terminal-objects-precategories.lagda.md +++ b/src/category-theory/terminal-objects-precategories.lagda.md @@ -1,4 +1,4 @@ -# Terminal object in a precategory +# Terminal objects in a precategory ```agda module category-theory.terminal-objects-precategories where @@ -12,9 +12,8 @@ open import category-theory.precategories open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.function-types +open import foundation.identity-types open import foundation.universe-levels - -open import foundation-core.identity-types ```
diff --git a/src/category-theory/yoneda-lemma-categories.lagda.md b/src/category-theory/yoneda-lemma-categories.lagda.md index 41533f11f5..87ebe92e2e 100644 --- a/src/category-theory/yoneda-lemma-categories.lagda.md +++ b/src/category-theory/yoneda-lemma-categories.lagda.md @@ -8,17 +8,13 @@ module category-theory.yoneda-lemma-categories where ```agda open import category-theory.categories -open import category-theory.functors-categories -open import category-theory.natural-transformations-functors-categories +open import category-theory.copresheaf-categories +open import category-theory.natural-transformations-functors-from-small-to-large-categories open import category-theory.representable-functors-categories open import category-theory.yoneda-lemma-precategories open import foundation.category-of-sets -open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.retractions -open import foundation.sections -open import foundation.sets open import foundation.universe-levels ``` @@ -28,61 +24,90 @@ open import foundation.universe-levels Given a [category](category-theory.categories.md) `C`, an object `c`, and a [functor](category-theory.functors-categories.md) `F` from `C` to the -[category of sets](foundation.category-of-sets.md), there is an -[equivalence](foundation-core.equivalences.md) between the +[category of sets](foundation.category-of-sets.md) + +```text + F : C → Set, +``` + +there is an [equivalence](foundation-core.equivalences.md) between the [set of natural transformations](category-theory.natural-transformations-functors-categories.md) from the functor [represented](category-theory.representable-functors-categories.md) by `c` to `F` and the [set](foundation-core.sets.md) `F c`. +```text + Nat(Hom(c , -) , F) ≃ F c +``` + More precisely, the **Yoneda lemma** asserts that the map from the type of natural transformations to the type `F c` defined by evaluating the component of the natural transformation at the object `c` at the identity arrow on `c` is an equivalence. -## Definition +## Theorem + +### The yoneda lemma into the large category of sets ```agda module _ - {l1 l2 : Level} (C : Category l1 l2) (c : obj-Category C) - (F : functor-Category C (Set-Category l2)) + {l1 l2 l3 : Level} (C : Category l1 l2) (c : obj-Category C) + (F : obj-copresheaf-Large-Category (precategory-Category C) l3) where - yoneda-evid-Category : - natural-transformation-Category - ( C) - ( Set-Category l2) - ( representable-functor-Category C c) - ( F) → - type-Set (obj-functor-Category C (Set-Category l2) F c) - yoneda-evid-Category = yoneda-evid-Precategory (precategory-Category C) c F - - yoneda-extension-Category : - type-Set (obj-functor-Category C (Set-Category l2) F c) → - natural-transformation-Category - C (Set-Category l2) (representable-functor-Category C c) F - yoneda-extension-Category = - yoneda-extension-Precategory (precategory-Category C) c F - - section-yoneda-evid-Category : - section yoneda-evid-Category - section-yoneda-evid-Category = - section-yoneda-evid-Precategory (precategory-Category C) c F - - retraction-yoneda-evid-Category : - retraction yoneda-evid-Category - retraction-yoneda-evid-Category = - retraction-yoneda-evid-Precategory (precategory-Category C) c F - - yoneda-lemma-Category : is-equiv yoneda-evid-Category - yoneda-lemma-Category = yoneda-lemma-Precategory (precategory-Category C) c F - - equiv-yoneda-lemma-Category : - ( natural-transformation-Category - ( C) - ( Set-Category l2) - ( representable-functor-Category C c) (F)) ≃ - ( type-Set (obj-functor-Category C (Set-Category l2) F c)) - pr1 equiv-yoneda-lemma-Category = yoneda-evid-Category - pr2 equiv-yoneda-lemma-Category = yoneda-lemma-Category + map-yoneda-Category : + hom-copresheaf-Large-Category + ( precategory-Category C) (representable-functor-Category C c) F → + section-copresheaf-Category (precategory-Category C) F c + map-yoneda-Category = + map-yoneda-Precategory (precategory-Category C) c F +``` + +The inverse to the Yoneda map: + +```agda + hom-family-extension-yoneda-Category : + (u : section-copresheaf-Category (precategory-Category C) F c) → + hom-family-functor-Small-Large-Category + C Set-Large-Category (representable-functor-Category C c) F + hom-family-extension-yoneda-Category = + hom-family-extension-yoneda-Precategory (precategory-Category C) c F + + naturality-extension-yoneda-Category : + (u : section-copresheaf-Category (precategory-Category C) F c) → + is-natural-transformation-Small-Large-Category + C Set-Large-Category (representable-functor-Category C c) F + ( hom-family-extension-yoneda-Category u) + naturality-extension-yoneda-Category = + naturality-extension-yoneda-Precategory (precategory-Category C) c F + + extension-yoneda-Category : + section-copresheaf-Category (precategory-Category C) F c → + hom-copresheaf-Large-Category + ( precategory-Category C) (representable-functor-Category C c) F + extension-yoneda-Category = + extension-yoneda-Precategory (precategory-Category C) c F + + lemma-yoneda-Category : is-equiv map-yoneda-Category + lemma-yoneda-Category = lemma-yoneda-Precategory (precategory-Category C) c F + + equiv-lemma-yoneda-Category : + hom-copresheaf-Large-Category + ( precategory-Category C) (representable-functor-Category C c) F ≃ + section-copresheaf-Category (precategory-Category C) F c + equiv-lemma-yoneda-Category = + equiv-lemma-yoneda-Precategory (precategory-Category C) c F ``` + +## See also + +- [Presheaf categories](category-theory.presheaf-categories.md) + +## External links + +- [The Yoneda embedding](https://1lab.dev/Cat.Functor.Hom.html#the-yoneda-embedding) + at 1lab +- [Yoneda lemma](https://ncatlab.org/nlab/show/Yoneda+lemma) at nlab +- [The Yoneda lemma](https://www.math3ma.com/blog/the-yoneda-lemma) at Math3ma +- [Yoneda lemma](https://en.wikipedia.org/wiki/Yoneda_lemma) at Wikipedia +- [Yoneda lemma](https://www.wikidata.org/wiki/Q320577) at Wikidata diff --git a/src/category-theory/yoneda-lemma-precategories.lagda.md b/src/category-theory/yoneda-lemma-precategories.lagda.md index 1c388211fc..e0e445ac2f 100644 --- a/src/category-theory/yoneda-lemma-precategories.lagda.md +++ b/src/category-theory/yoneda-lemma-precategories.lagda.md @@ -7,9 +7,13 @@ module category-theory.yoneda-lemma-precategories where
Imports ```agda +open import category-theory.copresheaf-categories +open import category-theory.functors-from-small-to-large-precategories open import category-theory.functors-precategories +open import category-theory.natural-transformations-functors-from-small-to-large-precategories open import category-theory.natural-transformations-functors-precategories open import category-theory.precategories +open import category-theory.presheaf-categories open import category-theory.representable-functors-precategories open import foundation.action-on-identifications-functions @@ -17,6 +21,8 @@ open import foundation.category-of-sets open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.function-types +open import foundation.homotopies open import foundation.identity-types open import foundation.retractions open import foundation.sections @@ -31,87 +37,136 @@ open import foundation.universe-levels Given a [precategory](category-theory.precategories.md) `C`, an object `c`, and a [functor](category-theory.functors-precategories.md) `F` from `C` to the -[precategory of sets](foundation.category-of-sets.md), there is an -[equivalence](foundation-core.equivalences.md) between the +[category of sets](foundation.category-of-sets.md) + +```text + F : C → Set, +``` + +there is an [equivalence](foundation-core.equivalences.md) between the [set of natural transformations](category-theory.natural-transformations-functors-precategories.md) from the functor [represented](category-theory.representable-functors-precategories.md) by `c` to `F` and the [set](foundation-core.sets.md) `F c`. +```text + Nat(Hom(c , -) , F) ≃ F c +``` + More precisely, the **Yoneda lemma** asserts that the map from the type of natural transformations to the type `F c` defined by evaluating the component of the natural transformation at the object `c` at the identity arrow on `c` is an equivalence. -## Definition +## Theorem + +### The yoneda lemma into the large category of sets ```agda module _ - {l1 l2 : Level} (C : Precategory l1 l2) (c : obj-Precategory C) - (F : functor-Precategory C (Set-Precategory l2)) + {l1 l2 l3 : Level} (C : Precategory l1 l2) (c : obj-Precategory C) + (F : obj-copresheaf-Large-Category C l3) where - yoneda-evid-Precategory : - natural-transformation-Precategory - ( C) - ( Set-Precategory l2) - ( representable-functor-Precategory C c) - ( F) → - type-Set (obj-functor-Precategory C (Set-Precategory l2) F c) - yoneda-evid-Precategory α = - hom-family-natural-transformation-Precategory + map-yoneda-Precategory : + hom-copresheaf-Large-Category C (representable-functor-Precategory C c) F → + section-copresheaf-Category C F c + map-yoneda-Precategory σ = + hom-family-natural-transformation-Small-Large-Precategory ( C) - ( Set-Precategory l2) + ( Set-Large-Precategory) ( representable-functor-Precategory C c) ( F) - ( α) + ( σ) ( c) ( id-hom-Precategory C) +``` - yoneda-extension-Precategory : - type-Set (obj-functor-Precategory C (Set-Precategory l2) F c) → - natural-transformation-Precategory - C (Set-Precategory l2) (representable-functor-Precategory C c) F - pr1 (yoneda-extension-Precategory u) x f = - hom-functor-Precategory C (Set-Precategory l2) F f u - pr2 (yoneda-extension-Precategory u) g = +The inverse to the Yoneda map: + +```agda + hom-family-extension-yoneda-Precategory : + (u : section-copresheaf-Category C F c) → + hom-family-functor-Small-Large-Precategory + C Set-Large-Precategory (representable-functor-Precategory C c) F + hom-family-extension-yoneda-Precategory u x f = + hom-functor-Small-Large-Precategory C Set-Large-Precategory F f u + + naturality-extension-yoneda-Precategory : + (u : section-copresheaf-Category C F c) → + is-natural-transformation-Small-Large-Precategory + C Set-Large-Precategory (representable-functor-Precategory C c) F + ( hom-family-extension-yoneda-Precategory u) + naturality-extension-yoneda-Precategory u g = eq-htpy ( λ f → htpy-eq ( inv - ( preserves-comp-functor-Precategory C (Set-Precategory l2) F g f)) + ( preserves-comp-functor-Small-Large-Precategory + C Set-Large-Precategory F g f)) ( u)) - section-yoneda-evid-Precategory : - section yoneda-evid-Precategory - pr1 section-yoneda-evid-Precategory = yoneda-extension-Precategory - pr2 section-yoneda-evid-Precategory = - htpy-eq (preserves-id-functor-Precategory C (Set-Precategory l2) F c) + extension-yoneda-Precategory : + section-copresheaf-Category C F c → + hom-copresheaf-Large-Category C (representable-functor-Precategory C c) F + pr1 (extension-yoneda-Precategory u) = + hom-family-extension-yoneda-Precategory u + pr2 (extension-yoneda-Precategory u) = + naturality-extension-yoneda-Precategory u +``` - retraction-yoneda-evid-Precategory : - retraction yoneda-evid-Precategory - pr1 retraction-yoneda-evid-Precategory = yoneda-extension-Precategory - pr2 retraction-yoneda-evid-Precategory α = +The inverse is an inverse: + +```agda + is-section-extension-yoneda-Precategory : + ( map-yoneda-Precategory ∘ + extension-yoneda-Precategory) ~ + id + is-section-extension-yoneda-Precategory = + htpy-eq + ( preserves-id-functor-Small-Large-Precategory + C Set-Large-Precategory F c) + + is-retraction-extension-yoneda-Precategory : + ( extension-yoneda-Precategory ∘ + map-yoneda-Precategory) ~ + id + is-retraction-extension-yoneda-Precategory σ = eq-type-subtype - ( is-natural-transformation-prop-Precategory - ( C) (Set-Precategory l2) (representable-functor-Precategory C c) F) + ( is-natural-transformation-prop-Small-Large-Precategory + ( C) Set-Large-Precategory (representable-functor-Precategory C c) F) ( eq-htpy ( λ x → eq-htpy ( λ f → ( htpy-eq - ( (pr2 α) f) - ( (id-hom-Precategory C))) ∙ - ( ap (pr1 α x) (right-unit-law-comp-hom-Precategory C f))))) - - yoneda-lemma-Precategory : is-equiv yoneda-evid-Precategory - pr1 yoneda-lemma-Precategory = section-yoneda-evid-Precategory - pr2 yoneda-lemma-Precategory = retraction-yoneda-evid-Precategory - - equiv-yoneda-lemma-Precategory : - ( natural-transformation-Precategory C (Set-Precategory l2) - ( representable-functor-Precategory C c) (F)) ≃ - ( type-Set (obj-functor-Precategory C (Set-Precategory l2) F c)) - pr1 equiv-yoneda-lemma-Precategory = yoneda-evid-Precategory - pr2 equiv-yoneda-lemma-Precategory = yoneda-lemma-Precategory + ( pr2 σ f) + ( id-hom-Precategory C)) ∙ + ( ap (pr1 σ x) (right-unit-law-comp-hom-Precategory C f))))) + + lemma-yoneda-Precategory : is-equiv map-yoneda-Precategory + lemma-yoneda-Precategory = + is-equiv-is-invertible + ( extension-yoneda-Precategory) + ( is-section-extension-yoneda-Precategory) + ( is-retraction-extension-yoneda-Precategory) + + equiv-lemma-yoneda-Precategory : + hom-copresheaf-Large-Category C (representable-functor-Precategory C c) F ≃ + section-copresheaf-Category C F c + pr1 equiv-lemma-yoneda-Precategory = map-yoneda-Precategory + pr2 equiv-lemma-yoneda-Precategory = lemma-yoneda-Precategory ``` + +## See also + +- [Presheaf categories](category-theory.presheaf-categories.md) + +## External links + +- [The Yoneda embedding](https://1lab.dev/Cat.Functor.Hom.html#the-yoneda-embedding) + at 1lab +- [Yoneda lemma](https://ncatlab.org/nlab/show/Yoneda+lemma) at nlab +- [The Yoneda lemma](https://www.math3ma.com/blog/the-yoneda-lemma) at Math3ma +- [Yoneda lemma](https://en.wikipedia.org/wiki/Yoneda_lemma) at Wikipedia +- [Yoneda lemma](https://www.wikidata.org/wiki/Q320577) at Wikidata diff --git a/src/elementary-number-theory/peano-arithmetic.lagda.md b/src/elementary-number-theory/peano-arithmetic.lagda.md index cb43d0bc30..ef15756228 100644 --- a/src/elementary-number-theory/peano-arithmetic.lagda.md +++ b/src/elementary-number-theory/peano-arithmetic.lagda.md @@ -41,7 +41,8 @@ peano-1-ℕ = zero-ℕ ## Peano's 2nd axiom -The identity relation on the natural numbers is reflexive. +The identity relation on the natural numbers is reflexive. I.e. for every +natural number `x`, it is true that `x ＝ x`. ```agda peano-axiom-2 : {l : Level} → UU l → UU l @@ -53,7 +54,8 @@ peano-2-ℕ x = refl ### Peano's 3rd axiom -The identity relation on the natural numbers is symmetric. +The identity relation on the natural numbers is symmetric. I.e. if `x ＝ y`, +then `y ＝ x`. ```agda peano-axiom-3 : {l : Level} → UU l → UU l @@ -65,7 +67,8 @@ peano-3-ℕ x y = inv ### Peano's 4th axiom -The identity relation on the natural numbers is transitive. +The identity relation on the natural numbers is transitive. I.e. if `y ＝ z` and +`x ＝ y`, then `x ＝ z`. ```agda peano-axiom-4 : {l : Level} → UU l → UU l @@ -80,7 +83,8 @@ peano-4-ℕ x y z = concat' x The 5th axiom of peano's arithmetic states that for every `x` and `y`, if `x ＝ y` and `y` is a natural number, then `x` is a natural number. This axiom does not make sense in type theory, as every element by definition lives in a -specified type. +specified type. To even pose the question of whether two elements are equal, we +must already know that they are elements of the same type. ### Peano's 6th axiom diff --git a/src/foundation-core/1-types.lagda.md b/src/foundation-core/1-types.lagda.md index f38795650c..79ac7bd491 100644 --- a/src/foundation-core/1-types.lagda.md +++ b/src/foundation-core/1-types.lagda.md @@ -7,13 +7,15 @@ module foundation-core.1-types where
Imports ```agda +open import foundation.contractible-types open import foundation.dependent-pair-types +open import foundation.truncated-types open import foundation.universe-levels open import foundation-core.equivalences open import foundation-core.identity-types +open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.truncated-types open import foundation-core.truncation-levels ``` @@ -52,9 +54,36 @@ pr2 (Id-Set X x y) = is-1-type-type-1-Type X x y ### Any set is a 1-type ```agda +abstract + is-1-type-is-set : + {l : Level} {A : UU l} → is-set A → is-1-type A + is-1-type-is-set = is-trunc-succ-is-trunc zero-𝕋 + 1-type-Set : {l : Level} → Set l → 1-Type l -1-type-Set A = truncated-type-succ-Truncated-Type zero-𝕋 A +1-type-Set = truncated-type-succ-Truncated-Type zero-𝕋 +``` + +### Any proposition is a 1-type + +```agda +abstract + is-1-type-is-prop : + {l : Level} {P : UU l} → is-prop P → is-1-type P + is-1-type-is-prop = is-trunc-iterated-succ-is-trunc neg-one-𝕋 2 + +1-type-Prop : + {l : Level} → Prop l → 1-Type l +1-type-Prop P = truncated-type-iterated-succ-Truncated-Type neg-one-𝕋 2 P +``` + +### Any contractible type is a 1-type + +```agda +abstract + is-1-type-is-contr : + {l : Level} {A : UU l} → is-contr A → is-1-type A + is-1-type-is-contr = is-trunc-is-contr one-𝕋 ``` ### The 1-types are closed under equivalences diff --git a/src/foundation-core/equality-dependent-pair-types.lagda.md b/src/foundation-core/equality-dependent-pair-types.lagda.md index 4b812e58ea..9cf8a462d2 100644 --- a/src/foundation-core/equality-dependent-pair-types.lagda.md +++ b/src/foundation-core/equality-dependent-pair-types.lagda.md @@ -60,17 +60,17 @@ module _ eq-pair-Σ' : {s t : Σ A B} → Eq-Σ s t → s ＝ t eq-pair-Σ' p = eq-pair-Σ (pr1 p) (pr2 p) - eq-pair-Σ-eq-pr1 : + eq-pair-eq-pr1 : {x y : A} {s : B x} (p : x ＝ y) → (x , s) ＝ (y , tr B p s) - eq-pair-Σ-eq-pr1 refl = refl + eq-pair-eq-pr1 refl = refl - eq-pair-Σ-eq-pr1' : + eq-pair-eq-pr1' : {x y : A} {t : B y} (p : x ＝ y) → (x , tr B (inv p) t) ＝ (y , t) - eq-pair-Σ-eq-pr1' refl = refl + eq-pair-eq-pr1' refl = refl - eq-pair-Σ-eq-pr2 : + eq-pair-eq-pr2 : {x : A} {s t : B x} → s ＝ t → (x , s) ＝ (x , t) - eq-pair-Σ-eq-pr2 {x} = ap {B = Σ A B} (pair x) + eq-pair-eq-pr2 {x} = ap {B = Σ A B} (pair x) is-retraction-pair-eq-Σ : (s t : Σ A B) → diff --git a/src/foundation-core/identity-types.lagda.md b/src/foundation-core/identity-types.lagda.md index 5fd87e8523..236c8d76ea 100644 --- a/src/foundation-core/identity-types.lagda.md +++ b/src/foundation-core/identity-types.lagda.md @@ -132,24 +132,24 @@ module _ ((p ∙ q) ∙ r) ＝ (p ∙ (q ∙ r)) assoc refl q r = refl - left-unit : {x y : A} {p : x ＝ y} → (refl ∙ p) ＝ p + left-unit : {x y : A} {p : x ＝ y} → refl ∙ p ＝ p left-unit = refl - right-unit : {x y : A} {p : x ＝ y} → (p ∙ refl) ＝ p + right-unit : {x y : A} {p : x ＝ y} → p ∙ refl ＝ p right-unit {p = refl} = refl - left-inv : {x y : A} (p : x ＝ y) → ((inv p) ∙ p) ＝ refl + left-inv : {x y : A} (p : x ＝ y) → inv p ∙ p ＝ refl left-inv refl = refl - right-inv : {x y : A} (p : x ＝ y) → (p ∙ (inv p)) ＝ refl + right-inv : {x y : A} (p : x ＝ y) → p ∙ (inv p) ＝ refl right-inv refl = refl - inv-inv : {x y : A} (p : x ＝ y) → (inv (inv p)) ＝ p + inv-inv : {x y : A} (p : x ＝ y) → inv (inv p) ＝ p inv-inv refl = refl distributive-inv-concat : {x y : A} (p : x ＝ y) {z : A} (q : y ＝ z) → - (inv (p ∙ q)) ＝ ((inv q) ∙ (inv p)) + inv (p ∙ q) ＝ inv q ∙ inv p distributive-inv-concat refl refl = refl ``` diff --git a/src/foundation-core/path-split-maps.lagda.md b/src/foundation-core/path-split-maps.lagda.md index 126b395ea6..66ae437fa4 100644 --- a/src/foundation-core/path-split-maps.lagda.md +++ b/src/foundation-core/path-split-maps.lagda.md @@ -90,8 +90,7 @@ module _ 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), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1308.0729)) + [arXiv:1308.0729](https://arxiv.org/abs/1308.0729)) 2. Mike Shulman, _Universal properties without function extensionality_ (November 2014) ([HoTT Blog](https://homotopytypetheory.org/2014/11/02/universal-properties-without-function-extensionality/)) diff --git a/src/foundation-core/propositions.lagda.md b/src/foundation-core/propositions.lagda.md index 1b394e0c5a..a7f898320f 100644 --- a/src/foundation-core/propositions.lagda.md +++ b/src/foundation-core/propositions.lagda.md @@ -270,229 +270,6 @@ pr1 (Π-Prop' A P) = type-Π-Prop' A P pr2 (Π-Prop' A P) = is-prop-Π' (λ x → is-prop-type-Prop (P x)) ``` -For convenience, we also record repeated applications of the above. - -#### Higher order products of families of propositions are propositions - -```agda -is-prop-Π² : - {l1 l2 l3 : Level} - {A1 : UU l1} {A2 : A1 → UU l2} {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} → - ((x1 : A1) (x2 : A2 x1) → is-prop (A3 x1 x2)) → - is-prop ((x1 : A1) (x2 : A2 x1) → A3 x1 x2) -is-prop-Π² H = is-prop-Π (is-prop-Π ∘ H) - -is-prop-Π³ : - {l1 l2 l3 l4 : Level} - {A1 : UU l1} - {A2 : A1 → UU l2} - {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} - {A4 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → UU l4} → - ((x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → is-prop (A4 x1 x2 x3)) → - is-prop ((x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → A4 x1 x2 x3) -is-prop-Π³ H = is-prop-Π (is-prop-Π² ∘ H) - -is-prop-Π⁴ : - {l1 l2 l3 l4 l5 : Level} - {A1 : UU l1} - {A2 : A1 → UU l2} - {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} - {A4 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → UU l4} - {A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5} → - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → - is-prop (A5 x1 x2 x3 x4)) → - is-prop - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → - A5 x1 x2 x3 x4) -is-prop-Π⁴ H = is-prop-Π (is-prop-Π³ ∘ H) - -is-prop-Π⁵ : - {l1 l2 l3 l4 l5 l6 : Level} - {A1 : UU l1} - {A2 : A1 → UU l2} - {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} - {A4 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → UU l4} - {A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5} → - {A6 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) → - UU l6} → - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) → - is-prop (A6 x1 x2 x3 x4 x5)) → - is-prop - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) → - A6 x1 x2 x3 x4 x5) -is-prop-Π⁵ H = is-prop-Π (is-prop-Π⁴ ∘ H) - -is-prop-Π⁶ : - {l1 l2 l3 l4 l5 l6 l7 : Level} - {A1 : UU l1} - {A2 : A1 → UU l2} - {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} - {A4 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → UU l4} - {A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5} → - {A6 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) → UU l6} → - {A7 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7} → - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → - is-prop (A7 x1 x2 x3 x4 x5 x6)) → - is-prop - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → - A7 x1 x2 x3 x4 x5 x6) -is-prop-Π⁶ H = is-prop-Π (is-prop-Π⁵ ∘ H) - -is-prop-Π⁷ : - {l1 l2 l3 l4 l5 l6 l7 l8 : Level} - {A1 : UU l1} - {A2 : A1 → UU l2} - {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} - {A4 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → UU l4} - {A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5} → - {A6 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) → UU l6} → - {A7 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7} → - {A8 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8} → - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) → - is-prop (A8 x1 x2 x3 x4 x5 x6 x7)) → - is-prop - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) → - A8 x1 x2 x3 x4 x5 x6 x7) -is-prop-Π⁷ H = is-prop-Π (is-prop-Π⁶ ∘ H) - -is-prop-Π⁸ : - {l1 l2 l3 l4 l5 l6 l7 l8 l9 : Level} - {A1 : UU l1} - {A2 : A1 → UU l2} - {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} - {A4 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → UU l4} - {A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5} - {A6 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) → UU l6} → - {A7 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7} - {A8 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8} - {A9 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) → UU l9} → - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) → - is-prop (A9 x1 x2 x3 x4 x5 x6 x7 x8)) → - is-prop - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) → - A9 x1 x2 x3 x4 x5 x6 x7 x8) -is-prop-Π⁸ H = is-prop-Π (is-prop-Π⁷ ∘ H) - -is-prop-Π⁹ : - {l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 : Level} - {A1 : UU l1} - {A2 : A1 → UU l2} - {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} - {A4 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → UU l4} - {A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5} - {A6 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) → UU l6} → - {A7 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7} - {A8 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8} - {A9 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) → UU l9} → - {A10 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) - (x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) → UU l10} → - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) - (x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) → - is-prop (A10 x1 x2 x3 x4 x5 x6 x7 x8 x9)) → - is-prop - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) - (x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) → - A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) -is-prop-Π⁹ H = is-prop-Π (is-prop-Π⁸ ∘ H) - -is-prop-Π¹⁰ : - {l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l11 : Level} - {A1 : UU l1} - {A2 : A1 → UU l2} - {A3 : (x1 : A1) (x2 : A2 x1) → UU l3} - {A4 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) → UU l4} - {A5 : (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) → UU l5} - {A6 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) → UU l6} → - {A7 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) → UU l7} - {A8 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) → UU l8} - {A9 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) → UU l9} → - {A10 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) - (x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) → UU l10} - {A11 : - (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) - (x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) → - UU l11} → - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) - (x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) → - is-prop (A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)) → - is-prop - ( (x1 : A1) (x2 : A2 x1) (x3 : A3 x1 x2) (x4 : A4 x1 x2 x3) - (x5 : A5 x1 x2 x3 x4) (x6 : A6 x1 x2 x3 x4 x5) - (x7 : A7 x1 x2 x3 x4 x5 x6) (x8 : A8 x1 x2 x3 x4 x5 x6 x7) - (x9 : A9 x1 x2 x3 x4 x5 x6 x7 x8) (x10 : A10 x1 x2 x3 x4 x5 x6 x7 x8 x9) → - A11 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) -is-prop-Π¹⁰ H = is-prop-Π (is-prop-Π⁹ ∘ H) -``` - ### The type of functions into a proposition is a proposition ```agda diff --git a/src/foundation-core/pullbacks.lagda.md b/src/foundation-core/pullbacks.lagda.md index 4573e9cffc..e047eee49c 100644 --- a/src/foundation-core/pullbacks.lagda.md +++ b/src/foundation-core/pullbacks.lagda.md @@ -298,8 +298,8 @@ inv-inv-map-commutative-standard-pullback : ( map-commutative-standard-pullback f g ∘ map-commutative-standard-pullback g f) ~ id inv-inv-map-commutative-standard-pullback f g x = - eq-pair-Σ-eq-pr2 - ( eq-pair-Σ-eq-pr2 + eq-pair-eq-pr2 + ( eq-pair-eq-pr2 ( inv-inv (coherence-square-standard-pullback x))) abstract @@ -522,7 +522,7 @@ triangle-map-prod-cone : ( gap (map-prod f f') (map-prod g g') (prod-cone f g f' g' c c')) ~ ( map-prod-cone f g f' g' ∘ map-prod (gap f g c) (gap f' g' c')) triangle-map-prod-cone f g c f' g' c' z = - eq-pair-Σ-eq-pr2 (eq-pair-Σ-eq-pr2 right-unit) + eq-pair-eq-pr2 (eq-pair-eq-pr2 right-unit) abstract is-equiv-map-prod-cone : @@ -688,8 +688,8 @@ module _ ( gap f g c ∘ map-equiv-total-fiber (pr1 c)) ~ ( tot (λ a → tot (λ b → inv)) ∘ tot (map-fiber-cone f g c)) square-tot-map-fiber-cone (.(vertical-map-cone f g c x) , x , refl) = - eq-pair-Σ-eq-pr2 - ( eq-pair-Σ-eq-pr2 + eq-pair-eq-pr2 + ( eq-pair-eq-pr2 ( inv (ap inv right-unit ∙ inv-inv (coherence-square-cone f g c x)))) abstract diff --git a/src/foundation-core/truncated-types.lagda.md b/src/foundation-core/truncated-types.lagda.md index bb510df533..942900bcbe 100644 --- a/src/foundation-core/truncated-types.lagda.md +++ b/src/foundation-core/truncated-types.lagda.md @@ -159,7 +159,7 @@ abstract is-trunc-is-equiv' k A f is-equiv-f ``` -### If a type embeds into a `k+1`-truncated type, then it is (k+1)-truncated +### If a type embeds into a `k+1`-truncated type, then it is `k+1`-truncated ```agda abstract diff --git a/src/foundation-core/truncation-levels.lagda.md b/src/foundation-core/truncation-levels.lagda.md index 00874df41c..ef346ace6e 100644 --- a/src/foundation-core/truncation-levels.lagda.md +++ b/src/foundation-core/truncation-levels.lagda.md @@ -14,16 +14,24 @@ open import foundation.universe-levels ## Idea -The type of truncation levels is a type similar to the type of natural numbers, -but starting the count at -2, so that sets have truncation level 0. +The type of **truncation levels** is a type similar to the type of +[natural numbers](elementary-number-theory.natural-numbers.md), but starting the +count at -2, so that [sets](foundation-core.sets.md) have +[truncation](foundation-core.truncated-types.md) level 0. -## Definition +## Definitions + +### The type of truncation levels ```agda data 𝕋 : UU lzero where neg-two-𝕋 : 𝕋 succ-𝕋 : 𝕋 → 𝕋 +``` +### Aliases for common truncation levels + +```agda neg-one-𝕋 : 𝕋 neg-one-𝕋 = succ-𝕋 neg-two-𝕋 diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md index 3568a211d2..33279e54bb 100644 --- a/src/foundation.lagda.md +++ b/src/foundation.lagda.md @@ -46,6 +46,7 @@ open import foundation.commuting-3-simplices-of-homotopies public open import foundation.commuting-3-simplices-of-maps public open import foundation.commuting-cubes-of-maps public open import foundation.commuting-hexagons-of-identifications public +open import foundation.commuting-pentagons-of-identifications public open import foundation.commuting-squares-of-homotopies public open import foundation.commuting-squares-of-identifications public open import foundation.commuting-squares-of-maps public diff --git a/src/foundation/1-types.lagda.md b/src/foundation/1-types.lagda.md index db5068c076..bba84b193b 100644 --- a/src/foundation/1-types.lagda.md +++ b/src/foundation/1-types.lagda.md @@ -11,15 +11,15 @@ open import foundation-core.1-types public ```agda open import foundation.dependent-pair-types open import foundation.subuniverses +open import foundation.truncated-types open import foundation.universe-levels -open import foundation-core.contractible-types +open import foundation-core.cartesian-product-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes open import foundation-core.torsorial-type-families -open import foundation-core.truncated-types open import foundation-core.truncation-levels ``` @@ -35,8 +35,17 @@ abstract is-1-type-Prop : {l : Level} → UU l → Prop l -pr1 (is-1-type-Prop A) = is-1-type A -pr2 (is-1-type-Prop A) = is-prop-is-1-type A +is-1-type-Prop = is-trunc-Prop one-𝕋 +``` + +### The type of all 1-types in a universe is a 2-type + +```agda +is-trunc-1-Type : {l : Level} → is-trunc two-𝕋 (1-Type l) +is-trunc-1-Type = is-trunc-Truncated-Type one-𝕋 + +1-Type-Truncated-Type : (l : Level) → Truncated-Type (lsuc l) two-𝕋 +1-Type-Truncated-Type l = Truncated-Type-Truncated-Type l one-𝕋 ``` ### Products of families of 1-types are 1-types @@ -66,19 +75,17 @@ pr2 (Π-1-Type' A B) = is-1-type-type-Π-1-Type' A B type-Π-1-Type : {l1 l2 : Level} (A : 1-Type l1) (B : type-1-Type A → 1-Type l2) → UU (l1 ⊔ l2) -type-Π-1-Type A B = type-Π-1-Type' (type-1-Type A) B +type-Π-1-Type A = type-Π-1-Type' (type-1-Type A) is-1-type-type-Π-1-Type : {l1 l2 : Level} (A : 1-Type l1) (B : type-1-Type A → 1-Type l2) → is-1-type (type-Π-1-Type A B) -is-1-type-type-Π-1-Type A B = - is-1-type-type-Π-1-Type' (type-1-Type A) B +is-1-type-type-Π-1-Type A = is-1-type-type-Π-1-Type' (type-1-Type A) Π-1-Type : {l1 l2 : Level} (A : 1-Type l1) (B : type-1-Type A → 1-Type l2) → 1-Type (l1 ⊔ l2) -pr1 (Π-1-Type A B) = type-Π-1-Type A B -pr2 (Π-1-Type A B) = is-1-type-type-Π-1-Type A B +Π-1-Type = Π-Truncated-Type one-𝕋 ``` ### The type of functions into a 1-type is a 1-type @@ -102,8 +109,35 @@ is-1-type-type-hom-1-Type A B = hom-1-Type : {l1 l2 : Level} (A : 1-Type l1) (B : 1-Type l2) → 1-Type (l1 ⊔ l2) -pr1 (hom-1-Type A B) = type-hom-1-Type A B -pr2 (hom-1-Type A B) = is-1-type-type-hom-1-Type A B +hom-1-Type = hom-Truncated-Type one-𝕋 +``` + +### 1-Types are closed under dependent pair types + +```agda +abstract + is-1-type-Σ : + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} → + is-1-type A → ((x : A) → is-1-type (B x)) → is-1-type (Σ A B) + is-1-type-Σ = is-trunc-Σ {k = one-𝕋} + +Σ-1-Type : + {l1 l2 : Level} (A : 1-Type l1) (B : pr1 A → 1-Type l2) → 1-Type (l1 ⊔ l2) +Σ-1-Type = Σ-Truncated-Type +``` + +### 1-Types are closed under cartesian product types + +```agda +abstract + is-1-type-prod : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → + is-1-type A → is-1-type B → is-1-type (A × B) + is-1-type-prod = is-trunc-prod one-𝕋 + +prod-1-Type : + {l1 l2 : Level} (A : 1-Type l1) (B : 1-Type l2) → 1-Type (l1 ⊔ l2) +prod-1-Type A B = Σ-1-Type A (λ x → B) ``` ### Subtypes of 1-types are 1-types diff --git a/src/foundation/binary-relations.lagda.md b/src/foundation/binary-relations.lagda.md index a8d57fb25c..e798315750 100644 --- a/src/foundation/binary-relations.lagda.md +++ b/src/foundation/binary-relations.lagda.md @@ -17,7 +17,6 @@ open import foundation.univalence open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.torsorial-type-families @@ -111,7 +110,7 @@ module _ is-prop-is-transitive-Relation-Prop : is-prop is-transitive-Relation-Prop is-prop-is-transitive-Relation-Prop = - is-prop-Π³ + is-prop-iterated-Π 3 ( λ x y z → is-prop-function-type ( is-prop-function-type (is-prop-type-Relation-Prop R x z))) diff --git a/src/foundation/cantor-schroder-bernstein-escardo.lagda.md b/src/foundation/cantor-schroder-bernstein-escardo.lagda.md index 1e526d26bd..4419efd46b 100644 --- a/src/foundation/cantor-schroder-bernstein-escardo.lagda.md +++ b/src/foundation/cantor-schroder-bernstein-escardo.lagda.md @@ -175,6 +175,6 @@ module _ - Martín H. Escardó, _The Cantor–Schröder–Bernstein Theorem for ∞-groupoids_, Journal of Homotopy and Related Structures, Volume 16, Issue 3, 2021 - ([arXiv:2002.07079](https://arxiv.org/abs/2002.07079),[DOI:10.1007](https://doi.org/10.1007/s40062-021-00284-6)) + ([arXiv:2002.07079](https://arxiv.org/abs/2002.07079),[DOI:10.1007/S40062-021-00284-6](https://doi.org/10.1007/s40062-021-00284-6)) - - diff --git a/src/foundation/category-of-sets.lagda.md b/src/foundation/category-of-sets.lagda.md index e8ee374ea4..efeaa27838 100644 --- a/src/foundation/category-of-sets.lagda.md +++ b/src/foundation/category-of-sets.lagda.md @@ -29,8 +29,10 @@ open import foundation-core.identity-types ## Idea -The **category of sets** consists of sets and functions. There is a category of -sets for each universe level, and there is a large category of sets. +The **category of [sets](foundation-core.sets.md)** consists of sets and +functions. There is a [category](category-theory.categories.md) of sets for each +universe level, and there is a +[large category](category-theory.large-categories.md) of sets. ## Definitions @@ -96,3 +98,19 @@ is-category-Set-Precategory : is-category-Set-Precategory l = is-category-Category (Set-Category l) ``` + +## Comments + +Since sets are equivalent to their set-truncations, the category of sets forms a +[full subprecategory](category-theory.full-large-subprecategories.md) of the +homotopy precategory of types. + +## See also + +- [Presheaf categories](category-theory.presheaf-categories.md) + +## External links + +- [Set](https://ncatlab.org/nlab/show/Set) at nlab +- [Category of sets](https://en.wikipedia.org/wiki/Category_of_sets) at + Wikipedia diff --git a/src/foundation/commuting-cubes-of-maps.lagda.md b/src/foundation/commuting-cubes-of-maps.lagda.md index 8f3fc76293..909c12b017 100644 --- a/src/foundation/commuting-cubes-of-maps.lagda.md +++ b/src/foundation/commuting-cubes-of-maps.lagda.md @@ -35,7 +35,7 @@ that the cube is presented as a lattice / | \ B' A C' |\ / \ /| - | \ ‌/ | + | \ / | |/ \ / \| B D' C' \ | / diff --git a/src/foundation/commuting-pentagons-of-identifications.lagda.md b/src/foundation/commuting-pentagons-of-identifications.lagda.md new file mode 100644 index 0000000000..eaf01a0440 --- /dev/null +++ b/src/foundation/commuting-pentagons-of-identifications.lagda.md @@ -0,0 +1,54 @@ +# Commuting pentagons of identifications + +```agda +module foundation.commuting-pentagons-of-identifications where +``` + +
Imports + +```agda +open import foundation.universe-levels + +open import foundation-core.identity-types +``` + +
+ +## Idea + +A pentagon of [identifications](foundation-core.identity-types.md) + +```text + top + x --- y +top-left / \ top-right + / \ + z w + \ / +bottom-left \ / bottom-right + v +``` + +is said to **commute** if there is an identification + +```text + top-left ∙ bottom-left ＝ (top ∙ top-right) ∙ bottom-right. +``` + +Such an identification is called a **coherence** of the pentagon. + +## Definition + +```agda +module _ + {l : Level} {A : UU l} {x y z w v : A} + where + + coherence-pentagon-identifications : + (top : x ＝ y) + (top-left : x ＝ z) (top-right : y ＝ w) + (bottom-left : z ＝ v) (bottom-right : w ＝ v) → UU l + coherence-pentagon-identifications + top top-left top-right bottom-left bottom-right = + top-left ∙ bottom-left ＝ (top ∙ top-right) ∙ bottom-right +``` diff --git a/src/foundation/decidable-embeddings.lagda.md b/src/foundation/decidable-embeddings.lagda.md index d13990b6db..2607d8a258 100644 --- a/src/foundation/decidable-embeddings.lagda.md +++ b/src/foundation/decidable-embeddings.lagda.md @@ -222,7 +222,7 @@ abstract is-prop-is-inhabited ( λ H → is-prop-prod - ( is-prop-is-emb f) + ( is-property-is-emb f) ( is-prop-Π ( λ y → is-prop-is-decidable (is-prop-map-is-emb (pr1 H) y)))) ``` diff --git a/src/foundation/decidable-equality.lagda.md b/src/foundation/decidable-equality.lagda.md index c61914611a..aba53c004a 100644 --- a/src/foundation/decidable-equality.lagda.md +++ b/src/foundation/decidable-equality.lagda.md @@ -35,8 +35,8 @@ open import foundation-core.transport-along-identifications ## Definition -A type `A` is said to have decidable equality if `Id x y` is a decidable type -for every `x y : A`. +A type `A` is said to have **decidable equality** if `x ＝ y` is a +[decidable type](foundation.decidable-types.md) for every `x y : A`. ```agda has-decidable-equality : {l : Level} (A : UU l) → UU l @@ -274,7 +274,7 @@ abstract ( is-set-has-decidable-equality dA (pr1 t) x))) ``` -### If B is a family of types with decidable equality, the total space has decidable equality, and B has a section, then the base type has decidable equality +### If `B` is a family of types with decidable equality, the total space has decidable equality, and `B` has a section, then the base type has decidable equality ```agda abstract diff --git a/src/foundation/double-negation-modality.lagda.md b/src/foundation/double-negation-modality.lagda.md index c62d92e0b0..2bd03a7af5 100644 --- a/src/foundation/double-negation-modality.lagda.md +++ b/src/foundation/double-negation-modality.lagda.md @@ -69,4 +69,4 @@ This proof follows Example 1.9 in _Modalities in homotopy type theory_. - Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) diff --git a/src/foundation/embeddings.lagda.md b/src/foundation/embeddings.lagda.md index 5ee493c330..ca25414087 100644 --- a/src/foundation/embeddings.lagda.md +++ b/src/foundation/embeddings.lagda.md @@ -44,13 +44,13 @@ module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} where - is-prop-is-emb : (f : A → B) → is-prop (is-emb f) - is-prop-is-emb f = + is-property-is-emb : (f : A → B) → is-prop (is-emb f) + is-property-is-emb f = is-prop-Π (λ x → is-prop-Π (λ y → is-property-is-equiv (ap f))) is-emb-Prop : (A → B) → Prop (l1 ⊔ l2) pr1 (is-emb-Prop f) = is-emb f - pr2 (is-emb-Prop f) = is-prop-is-emb f + pr2 (is-emb-Prop f) = is-property-is-emb f ``` ### Embeddings are closed under homotopies diff --git a/src/foundation/equality-dependent-function-types.lagda.md b/src/foundation/equality-dependent-function-types.lagda.md index 963b0063ed..bd8b9c4386 100644 --- a/src/foundation/equality-dependent-function-types.lagda.md +++ b/src/foundation/equality-dependent-function-types.lagda.md @@ -15,7 +15,6 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.equivalences -open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.torsorial-type-families diff --git a/src/foundation/functoriality-fibers-of-maps.lagda.md b/src/foundation/functoriality-fibers-of-maps.lagda.md index 96c0770060..48b82c08ac 100644 --- a/src/foundation/functoriality-fibers-of-maps.lagda.md +++ b/src/foundation/functoriality-fibers-of-maps.lagda.md @@ -65,7 +65,7 @@ module _ ( map-fiber-cone j h c (i x) ∘ map-fiber-cone i (pr1 c) d x) map-fiber-pasting-horizontal-cone (g , q , K) (f , p , H) .(f a) (a , refl) = - eq-pair-Σ-eq-pr2 + eq-pair-eq-pr2 ( ( ap ( concat' (h (q (p a))) refl) ( distributive-inv-concat (ap j (H a)) (K (p a)))) ∙ @@ -97,7 +97,7 @@ module _ map-fiber-pasting-vertical-cone (p , q , H) (p' , q' , H') .(p (p' a)) ((.(p' a) , refl) , (a , refl)) = - eq-pair-Σ-eq-pr2 + eq-pair-eq-pr2 ( ( right-unit) ∙ ( distributive-inv-concat (H (p' a)) (ap g (H' a))) ∙ ( ap diff --git a/src/foundation/identity-types.lagda.md b/src/foundation/identity-types.lagda.md index 2d48de1898..b15cec9771 100644 --- a/src/foundation/identity-types.lagda.md +++ b/src/foundation/identity-types.lagda.md @@ -11,6 +11,7 @@ open import foundation-core.identity-types public ```agda open import foundation.action-on-identifications-functions open import foundation.binary-equivalences +open import foundation.commuting-pentagons-of-identifications open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.function-extensionality @@ -44,17 +45,17 @@ identifications in arbitrary types. ### The Mac Lane pentagon for identity types ```agda -Mac-Lane-pentagon : +mac-lane-pentagon : {l : Level} {A : UU l} {a b c d e : A} (p : a ＝ b) (q : b ＝ c) (r : c ＝ d) (s : d ＝ e) → - let α₁ = (ap (λ t → t ∙ s) (assoc p q r)) + let α₁ = (ap (_∙ s) (assoc p q r)) α₂ = (assoc p (q ∙ r) s) - α₃ = (ap (λ t → p ∙ t) (assoc q r s)) + α₃ = (ap (p ∙_) (assoc q r s)) α₄ = (assoc (p ∙ q) r s) α₅ = (assoc p q (r ∙ s)) in - ((α₁ ∙ α₂) ∙ α₃) ＝ (α₄ ∙ α₅) -Mac-Lane-pentagon refl refl refl refl = refl + coherence-pentagon-identifications α₁ α₄ α₂ α₅ α₃ +mac-lane-pentagon refl refl refl refl = refl ``` ### The groupoidal operations on identity types are equivalences diff --git a/src/foundation/locally-small-types.lagda.md b/src/foundation/locally-small-types.lagda.md index f16321383d..a4835728c8 100644 --- a/src/foundation/locally-small-types.lagda.md +++ b/src/foundation/locally-small-types.lagda.md @@ -244,8 +244,7 @@ is-locally-small-inhabited-subtype H = ## References - Egbert Rijke, Theorem 4.6 in _The join construction_, 2017 - ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1701.07538)) + ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538)) - Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, and Daniel R. Grayson, Section 2.19 of _Symmetry_ ([draft](https://unimath.github.io/SymmetryBook/book.pdf), diff --git a/src/foundation/multivariable-homotopies.lagda.md b/src/foundation/multivariable-homotopies.lagda.md index 780ebd4274..a5aaf2529f 100644 --- a/src/foundation/multivariable-homotopies.lagda.md +++ b/src/foundation/multivariable-homotopies.lagda.md @@ -11,11 +11,11 @@ open import foundation.telescopes public ```agda open import elementary-number-theory.natural-numbers +open import foundation.equivalences open import foundation.function-extensionality open import foundation.iterated-dependent-product-types open import foundation.universe-levels -open import foundation-core.equivalences open import foundation-core.functoriality-dependent-function-types open import foundation-core.identity-types ``` diff --git a/src/foundation/path-split-maps.lagda.md b/src/foundation/path-split-maps.lagda.md index 9397e307b2..2d2df1f55e 100644 --- a/src/foundation/path-split-maps.lagda.md +++ b/src/foundation/path-split-maps.lagda.md @@ -84,8 +84,7 @@ module _ 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), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1308.0729)) + [arXiv:1308.0729](https://arxiv.org/abs/1308.0729)) 2. Mike Shulman, _Universal properties without function extensionality_ (November 2014) ([HoTT Blog](https://homotopytypetheory.org/2014/11/02/universal-properties-without-function-extensionality/)) diff --git a/src/foundation/preidempotent-maps.lagda.md b/src/foundation/preidempotent-maps.lagda.md index 8599010dd5..0e3a918a15 100644 --- a/src/foundation/preidempotent-maps.lagda.md +++ b/src/foundation/preidempotent-maps.lagda.md @@ -48,7 +48,6 @@ is-prop-is-preidempotent-is-set is-set-A f = - Mike Shulman, _Idempotents in intensional type theory_, Logical Methods in Computer Science, Volume 12, Issue 3, 2017 - ([arXiv:1507.03634](https://arxiv.org/abs/1507.03634), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1507.03634)) + ([arXiv:1507.03634](https://arxiv.org/abs/1507.03634)) - Mike Shulman, _Splitting Idempotents_, diff --git a/src/foundation/propositions.lagda.md b/src/foundation/propositions.lagda.md index 87c24471d8..563e24e52f 100644 --- a/src/foundation/propositions.lagda.md +++ b/src/foundation/propositions.lagda.md @@ -11,6 +11,7 @@ open import foundation-core.propositions public ```agda open import foundation.contractible-types open import foundation.dependent-pair-types +open import foundation.embeddings open import foundation.universe-levels open import foundation-core.retractions @@ -45,3 +46,18 @@ module _ is-prop-retract-of : A retract-of B → is-prop B → is-prop A is-prop-retract-of = is-trunc-retract-of ``` + +### If a type embeds into a proposition, then it is a proposition + +```agda +abstract + is-prop-is-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → + is-emb f → is-prop B → is-prop A + is-prop-is-emb = is-trunc-is-emb neg-two-𝕋 + +abstract + is-prop-emb : + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A ↪ B) → is-prop B → is-prop A + is-prop-emb = is-trunc-emb neg-two-𝕋 +``` diff --git a/src/foundation/pullbacks.lagda.md b/src/foundation/pullbacks.lagda.md index 4be90935f3..679af1bdf8 100644 --- a/src/foundation/pullbacks.lagda.md +++ b/src/foundation/pullbacks.lagda.md @@ -99,8 +99,8 @@ triangle-map-standard-pullback-exponent : ( exponent-cone T f g c))) triangle-map-standard-pullback-exponent {A = A} {B} T f g c h = - eq-pair-Σ-eq-pr2 - ( eq-pair-Σ-eq-pr2 + eq-pair-eq-pr2 + ( eq-pair-eq-pr2 ( inv (is-section-eq-htpy (coherence-square-cone f g c ·r h)))) abstract @@ -186,7 +186,7 @@ cone-Id' : cone (const unit (A × A) t) (diagonal A) (pr1 t ＝ pr2 t) pr1 (cone-Id' {A = A} (x , y)) = const (x ＝ y) unit star pr1 (pr2 (cone-Id' {A = A} (x , y))) = const (x ＝ y) A x -pr2 (pr2 (cone-Id' {A = A} (x , y))) p = eq-pair-Σ-eq-pr2 (inv p) +pr2 (pr2 (cone-Id' {A = A} (x , y))) p = eq-pair-eq-pr2 (inv p) inv-gap-cone-Id' : {l : Level} {A : UU l} (t : A × A) → diff --git a/src/foundation/replacement.lagda.md b/src/foundation/replacement.lagda.md index 8283b21027..23bf1e829f 100644 --- a/src/foundation/replacement.lagda.md +++ b/src/foundation/replacement.lagda.md @@ -57,8 +57,7 @@ replacement' f = replacement f is-small' ## References - Egbert Rijke, Theorem 4.6 in _The join construction_, 2017 - ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1701.07538)) + ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538)) - Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, and Daniel R. Grayson, Section 2.19 of _Symmetry_ ([newest version](https://unimath.github.io/SymmetryBook/book.pdf), diff --git a/src/foundation/sets.lagda.md b/src/foundation/sets.lagda.md index 27bdece44f..a43ea60edd 100644 --- a/src/foundation/sets.lagda.md +++ b/src/foundation/sets.lagda.md @@ -13,6 +13,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.subuniverses open import foundation.truncated-types +open import foundation.truncation-levels open import foundation.universe-levels open import foundation-core.1-types @@ -24,7 +25,6 @@ open import foundation-core.identity-types open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.torsorial-type-families -open import foundation-core.truncation-levels ```
@@ -241,9 +241,25 @@ abstract ```agda module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} where + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where is-emb-is-prop-is-set : is-prop A → is-set B → {f : A → B} → is-emb f is-emb-is-prop-is-set is-prop-A is-set-B {f} = is-emb-is-prop-map (λ b → is-prop-Σ is-prop-A (λ a → is-set-B (f a) b)) ``` + +### Sets are `k+2`-truncated for any `k` + +```agda +is-trunc-is-set : + {l : Level} (k : 𝕋) {A : UU l} → is-set A → is-trunc (succ-𝕋 (succ-𝕋 k)) A +is-trunc-is-set neg-two-𝕋 is-set-A = is-set-A +is-trunc-is-set (succ-𝕋 k) is-set-A = + is-trunc-succ-is-trunc (succ-𝕋 (succ-𝕋 k)) (is-trunc-is-set k is-set-A) + +set-Truncated-Type : + {l : Level} (k : 𝕋) → Set l → Truncated-Type l (succ-𝕋 (succ-𝕋 k)) +pr1 (set-Truncated-Type k A) = type-Set A +pr2 (set-Truncated-Type k A) = is-trunc-is-set k (is-set-type-Set A) +``` diff --git a/src/foundation/surjective-maps.lagda.md b/src/foundation/surjective-maps.lagda.md index e51675cf33..84827e97d5 100644 --- a/src/foundation/surjective-maps.lagda.md +++ b/src/foundation/surjective-maps.lagda.md @@ -721,7 +721,7 @@ module _ ( ap ( concat' (g (i a)) (M (f a))) ( is-section-map-inv-is-equiv - ( K (i a) ((j (f a)))) + ( K (i a) (j (f a))) ( L a ∙ inv (M (f a))))) ∙ ( is-section-inv-concat' (g (i a)) (M (f a)) (L a))))) where diff --git a/src/foundation/transport-along-higher-identifications.lagda.md b/src/foundation/transport-along-higher-identifications.lagda.md index babae89b03..e4d361e0c1 100644 --- a/src/foundation/transport-along-higher-identifications.lagda.md +++ b/src/foundation/transport-along-higher-identifications.lagda.md @@ -9,7 +9,6 @@ module foundation.transport-along-higher-identifications where ```agda open import foundation.action-on-identifications-functions open import foundation.commuting-squares-of-identifications -open import foundation.dependent-pair-types open import foundation.homotopies open import foundation.identity-types open import foundation.path-algebra diff --git a/src/foundation/transport-along-identifications.lagda.md b/src/foundation/transport-along-identifications.lagda.md index 12cf640e41..134e8ff91d 100644 --- a/src/foundation/transport-along-identifications.lagda.md +++ b/src/foundation/transport-along-identifications.lagda.md @@ -10,17 +10,13 @@ open import foundation-core.transport-along-identifications public ```agda open import foundation.action-on-identifications-functions -open import foundation.commuting-squares-of-identifications open import foundation.dependent-pair-types open import foundation.homotopies -open import foundation.path-algebra -open import foundation.transport-along-higher-identifications open import foundation.universe-levels open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.identity-types -open import foundation-core.whiskering-homotopies ```
diff --git a/src/foundation/truncated-types.lagda.md b/src/foundation/truncated-types.lagda.md index 2043af39ba..b9385ca2b3 100644 --- a/src/foundation/truncated-types.lagda.md +++ b/src/foundation/truncated-types.lagda.md @@ -9,18 +9,19 @@ open import foundation-core.truncated-types public
Imports ```agda +open import elementary-number-theory.natural-numbers + open import foundation.dependent-pair-types open import foundation.subtype-identity-principle +open import foundation.truncation-levels open import foundation.univalence open import foundation.universe-levels -open import foundation-core.contractible-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.subtypes open import foundation-core.torsorial-type-families -open import foundation-core.truncation-levels ```
@@ -72,3 +73,23 @@ pr2 (Truncated-Type-Truncated-Type l k) = is-trunc-Truncated-Type k emb-type-Truncated-Type : (l : Level) (k : 𝕋) → Truncated-Type l k ↪ UU l emb-type-Truncated-Type l k = emb-subtype (is-trunc-Prop k) ``` + +### If a type is `k`-truncated, then it is `k+r`-truncated + +```agda +abstract + is-trunc-iterated-succ-is-trunc : + (k : 𝕋) (r : ℕ) {l : Level} {A : UU l} → + is-trunc k A → is-trunc (iterated-succ-𝕋' k r) A + is-trunc-iterated-succ-is-trunc k zero-ℕ is-trunc-A = is-trunc-A + is-trunc-iterated-succ-is-trunc k (succ-ℕ r) is-trunc-A = + is-trunc-iterated-succ-is-trunc (succ-𝕋 k) r + ( is-trunc-succ-is-trunc k is-trunc-A) + +truncated-type-iterated-succ-Truncated-Type : + (k : 𝕋) (r : ℕ) {l : Level} → + Truncated-Type l k → Truncated-Type l (iterated-succ-𝕋' k r) +pr1 (truncated-type-iterated-succ-Truncated-Type k r A) = type-Truncated-Type A +pr2 (truncated-type-iterated-succ-Truncated-Type k r A) = + is-trunc-iterated-succ-is-trunc k r (is-trunc-type-Truncated-Type A) +``` diff --git a/src/foundation/truncation-levels.lagda.md b/src/foundation/truncation-levels.lagda.md index d2421c55b4..9b45dff830 100644 --- a/src/foundation/truncation-levels.lagda.md +++ b/src/foundation/truncation-levels.lagda.md @@ -36,6 +36,14 @@ truncation-level-ℕ : ℕ → 𝕋 truncation-level-ℕ = succ-𝕋 ∘ truncation-level-minus-one-ℕ ``` +### The inclusion of double successors of truncation levels into the natural numbers + +```agda +nat-succ-succ-𝕋 : 𝕋 → ℕ +nat-succ-succ-𝕋 neg-two-𝕋 = zero-ℕ +nat-succ-succ-𝕋 (succ-𝕋 k) = succ-ℕ (nat-succ-succ-𝕋 k) +``` + ### Addition of truncation levels ```agda @@ -52,6 +60,21 @@ infixl 35 _+𝕋_ _+𝕋_ = add-𝕋 ``` +### Iterated successor functions on truncation levels + +Although we can define an addition operation on truncation levels, when it comes +to doing induction on them, it is more natural to speak in terms of an iterated +successor: + +```agda +iterated-succ-𝕋 : ℕ → 𝕋 → 𝕋 +iterated-succ-𝕋 zero-ℕ x = x +iterated-succ-𝕋 (succ-ℕ n) x = iterated-succ-𝕋 n (succ-𝕋 x) + +iterated-succ-𝕋' : 𝕋 → ℕ → 𝕋 +iterated-succ-𝕋' x n = iterated-succ-𝕋 n x +``` + ## Properties ### Unit laws for addition of truncation levels diff --git a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md index c3b789a3cc..4e4bbb8561 100644 --- a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md +++ b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md @@ -41,24 +41,22 @@ module _ where map-inv-left-unit-law-Σ-is-contr : B a → Σ A B - map-inv-left-unit-law-Σ-is-contr b = pair a b + pr1 (map-inv-left-unit-law-Σ-is-contr b) = a + pr2 (map-inv-left-unit-law-Σ-is-contr b) = b map-left-unit-law-Σ-is-contr : Σ A B → B a map-left-unit-law-Σ-is-contr = - ind-Σ - ( ind-singleton a C - ( λ x → B x → B a) - ( id)) + ind-Σ (ind-singleton a C (λ x → B x → B a) (id)) is-section-map-inv-left-unit-law-Σ-is-contr : - ( map-left-unit-law-Σ-is-contr ∘ map-inv-left-unit-law-Σ-is-contr) ~ id + map-left-unit-law-Σ-is-contr ∘ map-inv-left-unit-law-Σ-is-contr ~ id is-section-map-inv-left-unit-law-Σ-is-contr b = ap ( λ (f : B a → B a) → f b) ( compute-ind-singleton a C (λ x → B x → B a) id) is-retraction-map-inv-left-unit-law-Σ-is-contr : - ( map-inv-left-unit-law-Σ-is-contr ∘ map-left-unit-law-Σ-is-contr) ~ id + map-inv-left-unit-law-Σ-is-contr ∘ map-left-unit-law-Σ-is-contr ~ id is-retraction-map-inv-left-unit-law-Σ-is-contr = ind-Σ ( ind-singleton a C @@ -67,8 +65,8 @@ module _ Id ( ( map-inv-left-unit-law-Σ-is-contr ∘ map-left-unit-law-Σ-is-contr) - ( pair x y)) - ( pair x y)) + ( x , y)) + ( x , y)) ( λ y → ap ( map-inv-left-unit-law-Σ-is-contr) ( ap @@ -138,12 +136,12 @@ module _ is-section-map-inv-right-unit-law-Σ-is-contr : (H : (a : A) → is-contr (B a)) → - ( pr1 ∘ map-inv-right-unit-law-Σ-is-contr H) ~ id + pr1 ∘ map-inv-right-unit-law-Σ-is-contr H ~ id is-section-map-inv-right-unit-law-Σ-is-contr H = refl-htpy is-retraction-map-inv-right-unit-law-Σ-is-contr : (H : (a : A) → is-contr (B a)) → - ( map-inv-right-unit-law-Σ-is-contr H ∘ pr1) ~ id + map-inv-right-unit-law-Σ-is-contr H ∘ pr1 ~ id is-retraction-map-inv-right-unit-law-Σ-is-contr H (a , b) = eq-pair-Σ refl (eq-is-contr (H a)) @@ -173,23 +171,23 @@ module _ {l1 l2 l3 : Level} (A : UU l1) (B : A → UU l2) (C : Σ A B → UU l3) where - map-associative-Σ : Σ (Σ A B) C → Σ A (λ x → Σ (B x) (λ y → C (pair x y))) + map-associative-Σ : Σ (Σ A B) C → Σ A (λ x → Σ (B x) (λ y → C (x , y))) pr1 (map-associative-Σ ((x , y) , z)) = x pr1 (pr2 (map-associative-Σ ((x , y) , z))) = y pr2 (pr2 (map-associative-Σ ((x , y) , z))) = z - map-inv-associative-Σ : Σ A (λ x → Σ (B x) (λ y → C (pair x y))) → Σ (Σ A B) C + map-inv-associative-Σ : Σ A (λ x → Σ (B x) (λ y → C (x , y))) → Σ (Σ A B) C pr1 (pr1 (map-inv-associative-Σ (x , y , z))) = x pr2 (pr1 (map-inv-associative-Σ (x , y , z))) = y pr2 (map-inv-associative-Σ (x , y , z)) = z is-retraction-map-inv-associative-Σ : - (map-inv-associative-Σ ∘ map-associative-Σ) ~ id - is-retraction-map-inv-associative-Σ (pair (pair x y) z) = refl + map-inv-associative-Σ ∘ map-associative-Σ ~ id + is-retraction-map-inv-associative-Σ ((x , y) , z) = refl is-section-map-inv-associative-Σ : - (map-associative-Σ ∘ map-inv-associative-Σ) ~ id - is-section-map-inv-associative-Σ (pair x (pair y z)) = refl + map-associative-Σ ∘ map-inv-associative-Σ ~ id + is-section-map-inv-associative-Σ (x , (y , z)) = refl abstract is-equiv-map-associative-Σ : is-equiv map-associative-Σ @@ -199,11 +197,11 @@ module _ is-section-map-inv-associative-Σ is-retraction-map-inv-associative-Σ - associative-Σ : Σ (Σ A B) C ≃ Σ A (λ x → Σ (B x) (λ y → C (pair x y))) + associative-Σ : Σ (Σ A B) C ≃ Σ A (λ x → Σ (B x) (λ y → C (x , y))) pr1 associative-Σ = map-associative-Σ pr2 associative-Σ = is-equiv-map-associative-Σ - inv-associative-Σ : Σ A (λ x → Σ (B x) (λ y → C (pair x y))) ≃ Σ (Σ A B) C + inv-associative-Σ : Σ A (λ x → Σ (B x) (λ y → C (x , y))) ≃ Σ (Σ A B) C pr1 inv-associative-Σ = map-inv-associative-Σ pr2 inv-associative-Σ = is-equiv-is-invertible @@ -232,12 +230,12 @@ module _ pr2 (map-inv-associative-Σ' (x , y , z)) = z is-section-map-inv-associative-Σ' : - (map-associative-Σ' ∘ map-inv-associative-Σ') ~ id - is-section-map-inv-associative-Σ' (pair x (pair y z)) = refl + map-associative-Σ' ∘ map-inv-associative-Σ' ~ id + is-section-map-inv-associative-Σ' (x , (y , z)) = refl is-retraction-map-inv-associative-Σ' : - ( map-inv-associative-Σ' ∘ map-associative-Σ') ~ id - is-retraction-map-inv-associative-Σ' (pair (pair x y) z) = refl + map-inv-associative-Σ' ∘ map-associative-Σ' ~ id + is-retraction-map-inv-associative-Σ' ((x , y) , z) = refl is-equiv-map-associative-Σ' : is-equiv map-associative-Σ' is-equiv-map-associative-Σ' = @@ -265,7 +263,7 @@ module _ ```agda module _ - { l1 l2 l3 l4 : Level} { A : UU l1} {B : A → UU l2} {C : A → UU l3} + { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} ( D : (x : A) → B x → C x → UU l4) where @@ -286,12 +284,12 @@ module _ pr2 (pr2 (map-inv-interchange-Σ-Σ t)) = pr2 (pr2 t) is-section-map-inv-interchange-Σ-Σ : - ( map-interchange-Σ-Σ ∘ map-inv-interchange-Σ-Σ) ~ id - is-section-map-inv-interchange-Σ-Σ (pair (pair a c) (pair b d)) = refl + map-interchange-Σ-Σ ∘ map-inv-interchange-Σ-Σ ~ id + is-section-map-inv-interchange-Σ-Σ ((a , c) , (b , d)) = refl is-retraction-map-inv-interchange-Σ-Σ : - ( map-inv-interchange-Σ-Σ ∘ map-interchange-Σ-Σ) ~ id - is-retraction-map-inv-interchange-Σ-Σ (pair (pair a b) (pair c d)) = refl + map-inv-interchange-Σ-Σ ∘ map-interchange-Σ-Σ ~ id + is-retraction-map-inv-interchange-Σ-Σ ((a , b) , (c , d)) = refl abstract is-equiv-map-interchange-Σ-Σ : is-equiv map-interchange-Σ-Σ @@ -335,11 +333,11 @@ module _ pr2 (pr2 (map-inv-left-swap-Σ (b , a , c))) = c is-retraction-map-inv-left-swap-Σ : - (map-inv-left-swap-Σ ∘ map-left-swap-Σ) ~ id - is-retraction-map-inv-left-swap-Σ (pair a (pair b c)) = refl + map-inv-left-swap-Σ ∘ map-left-swap-Σ ~ id + is-retraction-map-inv-left-swap-Σ (a , (b , c)) = refl - is-section-map-inv-left-swap-Σ : (map-left-swap-Σ ∘ map-inv-left-swap-Σ) ~ id - is-section-map-inv-left-swap-Σ (pair b (pair a c)) = refl + is-section-map-inv-left-swap-Σ : map-left-swap-Σ ∘ map-inv-left-swap-Σ ~ id + is-section-map-inv-left-swap-Σ (b , (a , c)) = refl abstract is-equiv-map-left-swap-Σ : is-equiv map-left-swap-Σ @@ -372,12 +370,12 @@ module _ pr2 (map-inv-right-swap-Σ ((a , c) , b)) = c is-section-map-inv-right-swap-Σ : - (map-right-swap-Σ ∘ map-inv-right-swap-Σ) ~ id - is-section-map-inv-right-swap-Σ (pair (pair x y) z) = refl + map-right-swap-Σ ∘ map-inv-right-swap-Σ ~ id + is-section-map-inv-right-swap-Σ ((x , y) , z) = refl is-retraction-map-inv-right-swap-Σ : - (map-inv-right-swap-Σ ∘ map-right-swap-Σ) ~ id - is-retraction-map-inv-right-swap-Σ (pair (pair x z) y) = refl + map-inv-right-swap-Σ ∘ map-right-swap-Σ ~ id + is-retraction-map-inv-right-swap-Σ ((x , z) , y) = refl is-equiv-map-right-swap-Σ : is-equiv map-right-swap-Σ is-equiv-map-right-swap-Σ = diff --git a/src/group-theory/automorphism-groups.lagda.md b/src/group-theory/automorphism-groups.lagda.md index 6c7cc48feb..66a609eded 100644 --- a/src/group-theory/automorphism-groups.lagda.md +++ b/src/group-theory/automorphism-groups.lagda.md @@ -33,7 +33,8 @@ open import structured-types.pointed-types ## Idea -The automorphim group of `a : A` is the group of symmetries of `a` in `A`. +The **automorphim group** of `a : A` is the [group](group-theory.groups.md) of +symmetries of `a` in `A`. ## Definitions diff --git a/src/group-theory/concrete-monoids.lagda.md b/src/group-theory/concrete-monoids.lagda.md index e8c280734b..702c2b25d3 100644 --- a/src/group-theory/concrete-monoids.lagda.md +++ b/src/group-theory/concrete-monoids.lagda.md @@ -58,7 +58,7 @@ Given a monoid, we can define its associated concrete monoid. The type of objects is the [classifying type](group-theory.concrete-groups.md) of the [core](group-theory.cores-monoids.md) of the monoid. Moreover, we must take care in how we define the family of homomorphisms. They cannot simply be the constant -family, as [transporting](foundation.transport-along-identifications.md) along +family, as [transporting along](foundation.transport-along-identifications.md) an [invertible element](group-theory.invertible-elements-monoids.md) should correspond to multiplying by the element in the family. diff --git a/src/group-theory/torsors.lagda.md b/src/group-theory/torsors.lagda.md index 0a00390f75..5516169b25 100644 --- a/src/group-theory/torsors.lagda.md +++ b/src/group-theory/torsors.lagda.md @@ -9,7 +9,6 @@ module group-theory.torsors where ```agda open import foundation.0-connected-types open import foundation.action-on-identifications-functions -open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-types @@ -42,8 +41,9 @@ open import higher-group-theory.higher-groups ## Idea -A torsor of `G` is a group action wich merely equivalent to the principal group -action of `G`. +A **torsor** of `G` is a [group action](group-theory.group-actions.md) which is +[merely equivalent](foundation.mere-equivalences.md) to the +[principal group action](group-theory.principal-group-actions.md) of `G`. ## Definitions diff --git a/src/order-theory/large-posets.lagda.md b/src/order-theory/large-posets.lagda.md index ccd905d046..fd55dfa299 100644 --- a/src/order-theory/large-posets.lagda.md +++ b/src/order-theory/large-posets.lagda.md @@ -174,7 +174,7 @@ module _ is-large-category-Large-Poset {l} x y = is-equiv-is-prop ( is-set-type-Large-Poset P x y) - ( is-prop-iso-Precategory + ( is-prop-iso-is-prop-hom-Precategory ( precategory-Large-Poset l) ( is-prop-leq-Large-Poset P x y)) ( λ f → diff --git a/src/order-theory/posets.lagda.md b/src/order-theory/posets.lagda.md index 1a3c6eb992..8b9ee2314c 100644 --- a/src/order-theory/posets.lagda.md +++ b/src/order-theory/posets.lagda.md @@ -157,7 +157,8 @@ module _ is-category-precategory-Poset x y = is-equiv-is-prop ( is-set-type-Poset X x y) - ( is-prop-iso-Precategory precategory-Poset (is-prop-leq-Poset X x y)) + ( is-prop-iso-is-prop-hom-Precategory precategory-Poset + ( is-prop-leq-Poset X x y)) ( λ f → antisymmetric-leq-Poset X x y ( hom-iso-Precategory precategory-Poset f) @@ -169,8 +170,7 @@ module _ module _ {l1 l2 : Level} (C : Category l1 l2) - ( is-prop-hom-C : - (x y : obj-Category C) → is-prop (hom-Category C x y)) + (is-prop-hom-C : (x y : obj-Category C) → is-prop (hom-Category C x y)) where preorder-is-prop-hom-Category : Preorder l1 l2 diff --git a/src/orthogonal-factorization-systems/closed-modalities.lagda.md b/src/orthogonal-factorization-systems/closed-modalities.lagda.md index d8b6f9a452..d39203d1c0 100644 --- a/src/orthogonal-factorization-systems/closed-modalities.lagda.md +++ b/src/orthogonal-factorization-systems/closed-modalities.lagda.md @@ -122,4 +122,4 @@ module _ - Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) diff --git a/src/orthogonal-factorization-systems/higher-modalities.lagda.md b/src/orthogonal-factorization-systems/higher-modalities.lagda.md index 24b4ee3b68..26554e0958 100644 --- a/src/orthogonal-factorization-systems/higher-modalities.lagda.md +++ b/src/orthogonal-factorization-systems/higher-modalities.lagda.md @@ -354,4 +354,4 @@ The equivalent notions of - Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) diff --git a/src/orthogonal-factorization-systems/localizations-maps.lagda.md b/src/orthogonal-factorization-systems/localizations-maps.lagda.md index c904c96be9..afdb8a9d4b 100644 --- a/src/orthogonal-factorization-systems/localizations-maps.lagda.md +++ b/src/orthogonal-factorization-systems/localizations-maps.lagda.md @@ -92,7 +92,6 @@ It remains to construct a converse. 1. Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) 2. Egbert Rijke, _Classifying Types_ - ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435), - [doi:10.48550](https://doi.org/10.48550/arXiv.1906.09435)) + ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435)) diff --git a/src/orthogonal-factorization-systems/localizations-subuniverses.lagda.md b/src/orthogonal-factorization-systems/localizations-subuniverses.lagda.md index b48b595c6a..d5903a78da 100644 --- a/src/orthogonal-factorization-systems/localizations-subuniverses.lagda.md +++ b/src/orthogonal-factorization-systems/localizations-subuniverses.lagda.md @@ -119,7 +119,6 @@ This is Proposition 5.1.2 in _Classifying Types_, and remains to be formalized. 1. Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) 2. Egbert Rijke, _Classifying Types_ - ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435), - [doi:10.48550](https://doi.org/10.48550/arXiv.1906.09435)) + ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435)) diff --git a/src/orthogonal-factorization-systems/modal-operators.lagda.md b/src/orthogonal-factorization-systems/modal-operators.lagda.md index ea6b4b64dc..d178089800 100644 --- a/src/orthogonal-factorization-systems/modal-operators.lagda.md +++ b/src/orthogonal-factorization-systems/modal-operators.lagda.md @@ -147,4 +147,4 @@ module _ - Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) diff --git a/src/orthogonal-factorization-systems/orthogonal-factorization-systems.lagda.md b/src/orthogonal-factorization-systems/orthogonal-factorization-systems.lagda.md index 328d0a5873..e59c641fc3 100644 --- a/src/orthogonal-factorization-systems/orthogonal-factorization-systems.lagda.md +++ b/src/orthogonal-factorization-systems/orthogonal-factorization-systems.lagda.md @@ -189,4 +189,4 @@ This remains to be shown. - Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) diff --git a/src/orthogonal-factorization-systems/reflective-subuniverses.lagda.md b/src/orthogonal-factorization-systems/reflective-subuniverses.lagda.md index 49057ab99b..d2f588bdd1 100644 --- a/src/orthogonal-factorization-systems/reflective-subuniverses.lagda.md +++ b/src/orthogonal-factorization-systems/reflective-subuniverses.lagda.md @@ -187,12 +187,10 @@ module _ 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), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1308.0729)) + [arXiv:1308.0729](https://arxiv.org/abs/1308.0729)) 2. Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) 3. Egbert Rijke, _Classifying Types_ - ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435), - [doi:10.48550](https://doi.org/10.48550/arXiv.1906.09435)) + ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435)) diff --git a/src/orthogonal-factorization-systems/separated-types.lagda.md b/src/orthogonal-factorization-systems/separated-types.lagda.md index 7402a89313..d773fc900e 100644 --- a/src/orthogonal-factorization-systems/separated-types.lagda.md +++ b/src/orthogonal-factorization-systems/separated-types.lagda.md @@ -38,5 +38,4 @@ module _ ## References 1. Egbert Rijke, _Classifying Types_ - ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435), - [doi:10.48550](https://doi.org/10.48550/arXiv.1906.09435)) + ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435)) diff --git a/src/orthogonal-factorization-systems/sigma-closed-reflective-subuniverses.lagda.md b/src/orthogonal-factorization-systems/sigma-closed-reflective-subuniverses.lagda.md index 5b037b7ed8..1f845ce440 100644 --- a/src/orthogonal-factorization-systems/sigma-closed-reflective-subuniverses.lagda.md +++ b/src/orthogonal-factorization-systems/sigma-closed-reflective-subuniverses.lagda.md @@ -52,4 +52,4 @@ The equivalent notions of - Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) diff --git a/src/orthogonal-factorization-systems/stable-orthogonal-factorization-systems.lagda.md b/src/orthogonal-factorization-systems/stable-orthogonal-factorization-systems.lagda.md index fad1c15f29..da36de111e 100644 --- a/src/orthogonal-factorization-systems/stable-orthogonal-factorization-systems.lagda.md +++ b/src/orthogonal-factorization-systems/stable-orthogonal-factorization-systems.lagda.md @@ -50,4 +50,4 @@ The equivalent notions of - Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) diff --git a/src/orthogonal-factorization-systems/uniquely-eliminating-modalities.lagda.md b/src/orthogonal-factorization-systems/uniquely-eliminating-modalities.lagda.md index f01cf72205..76f7bdfd1c 100644 --- a/src/orthogonal-factorization-systems/uniquely-eliminating-modalities.lagda.md +++ b/src/orthogonal-factorization-systems/uniquely-eliminating-modalities.lagda.md @@ -203,4 +203,4 @@ The equivalent notions of - Egbert Rijke, Michael Shulman, Bas Spitters, _Modalities in homotopy type theory_, Logical Methods in Computer Science, Volume 16, Issue 1, 2020 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), - [doi:10.23638](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) + [DOI:10.23638/LMCS-16(1:2)2020](https://doi.org/10.23638/LMCS-16%281%3A2%292020)) diff --git a/src/set-theory/cumulative-hierarchy.lagda.md b/src/set-theory/cumulative-hierarchy.lagda.md index 7dae074724..95866ae5ba 100644 --- a/src/set-theory/cumulative-hierarchy.lagda.md +++ b/src/set-theory/cumulative-hierarchy.lagda.md @@ -764,8 +764,7 @@ needed. 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), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1308.0729)) + [arXiv:1308.0729](https://arxiv.org/abs/1308.0729)) 2. Tom de Jong, in collaboration with Nicolai Kraus, Fredrik Nordvall Forsberg and Chuangjie Xu. diff --git a/src/synthetic-homotopy-theory/joins-of-types.lagda.md b/src/synthetic-homotopy-theory/joins-of-types.lagda.md index b1bf34beb5..4da4766e1e 100644 --- a/src/synthetic-homotopy-theory/joins-of-types.lagda.md +++ b/src/synthetic-homotopy-theory/joins-of-types.lagda.md @@ -361,5 +361,4 @@ module _ ## References - Egbert Rijke, _The join construction_, 2017 - ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538), - [DOI:10.48550](https://doi.org/10.48550/arXiv.1701.07538)) + ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538))