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| # Postcomposition of pullbacks | ||
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| ```agda | ||
| module foundation.postcomposition-pullbacks where | ||
| ``` | ||
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| <details><summary>Imports</summary> | ||
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| ```agda | ||
| open import foundation.action-on-identifications-functions | ||
| open import foundation.cones-over-cospan-diagrams | ||
| open import foundation.coproduct-types | ||
| open import foundation.dependent-pair-types | ||
| open import foundation.equality-coproduct-types | ||
| open import foundation.function-extensionality | ||
| open import foundation.functoriality-coproduct-types | ||
| open import foundation.functoriality-function-types | ||
| open import foundation.identity-types | ||
| open import foundation.universe-levels | ||
| open import foundation.whiskering-homotopies-composition | ||
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| open import foundation-core.equality-dependent-pair-types | ||
| open import foundation-core.equivalences | ||
| open import foundation-core.function-types | ||
| open import foundation-core.functoriality-dependent-pair-types | ||
| open import foundation-core.homotopies | ||
| open import foundation-core.postcomposition-functions | ||
| open import foundation-core.pullbacks | ||
| open import foundation-core.retractions | ||
| open import foundation-core.sections | ||
| open import foundation-core.standard-pullbacks | ||
| open import foundation-core.universal-property-pullbacks | ||
| ``` | ||
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| </details> | ||
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| ## Idea | ||
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| Given a [pullback](foundation-core.pullbacks.md) square | ||
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| ```text | ||
| f' | ||
| C -------> B | ||
| | ⌟ | | ||
| g'| | g | ||
| ∨ ∨ | ||
| A -------> X | ||
| f | ||
| ``` | ||
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| then the exponentiated square given by | ||
| [postcomposition](foundation-core.postcomposition-functions.md) | ||
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| ```text | ||
| f' ∘ - | ||
| (T → C) ---------> (T → B) | ||
| | | | ||
| g' ∘ - | | g ∘ - | ||
| | | | ||
| ∨ ∨ | ||
| (T → A) ---------> (T → X) | ||
| f ∘ - | ||
| ``` | ||
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| is a pullback square for any type `T`. | ||
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| ## Definitions | ||
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| ### Postcomposition cones | ||
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| ```agda | ||
| postcomp-cone : | ||
| {l1 l2 l3 l4 l5 : Level} | ||
| {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} (T : UU l5) | ||
| (f : A → X) (g : B → X) (c : cone f g C) → | ||
| cone (postcomp T f) (postcomp T g) (T → C) | ||
| pr1 (postcomp-cone T f g c) h = vertical-map-cone f g c ∘ h | ||
| pr1 (pr2 (postcomp-cone T f g c)) h = horizontal-map-cone f g c ∘ h | ||
| pr2 (pr2 (postcomp-cone T f g c)) h = eq-htpy (coherence-square-cone f g c ·r h) | ||
| ``` | ||
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| ## Properties | ||
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| ### Standard pullbacks are closed under postcomposition exponentiation | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 l3 l4 : Level} | ||
| {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) | ||
| (T : UU l4) | ||
| where | ||
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| map-postcomp-cone-standard-pullback : | ||
| standard-pullback (postcomp T f) (postcomp T g) → cone f g T | ||
| map-postcomp-cone-standard-pullback = tot (λ _ → tot (λ _ → htpy-eq)) | ||
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| abstract | ||
| is-equiv-map-postcomp-cone-standard-pullback : | ||
| is-equiv map-postcomp-cone-standard-pullback | ||
| is-equiv-map-postcomp-cone-standard-pullback = | ||
| is-equiv-tot-is-fiberwise-equiv | ||
| ( λ p → is-equiv-tot-is-fiberwise-equiv (λ q → funext (f ∘ p) (g ∘ q))) | ||
| ``` | ||
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| ### The precomposition action on cones computes as the gap map of a postcomposition cone | ||
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| ```agda | ||
| triangle-map-postcomp-cone-standard-pullback : | ||
| {l1 l2 l3 l4 l5 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} | ||
| (T : UU l5) (f : A → X) (g : B → X) (c : cone f g C) → | ||
| cone-map f g c {T} ~ | ||
| map-postcomp-cone-standard-pullback f g T ∘ | ||
| gap (postcomp T f) (postcomp T g) (postcomp-cone T f g c) | ||
| triangle-map-postcomp-cone-standard-pullback T f g c h = | ||
| eq-pair-eq-fiber | ||
| ( eq-pair-eq-fiber | ||
| ( inv (is-section-eq-htpy (coherence-square-cone f g c ·r h)))) | ||
| ``` | ||
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| ### Pullbacks are closed under postcomposition exponentiation | ||
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| ```agda | ||
| abstract | ||
| is-pullback-postcomp-is-pullback : | ||
| {l1 l2 l3 l4 l5 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} | ||
| (f : A → X) (g : B → X) (c : cone f g C) → is-pullback f g c → | ||
| (T : UU l5) → | ||
| is-pullback (postcomp T f) (postcomp T g) (postcomp-cone T f g c) | ||
| is-pullback-postcomp-is-pullback f g c is-pb-c T = | ||
| is-equiv-top-map-triangle | ||
| ( cone-map f g c) | ||
| ( map-postcomp-cone-standard-pullback f g T) | ||
| ( gap (f ∘_) (g ∘_) (postcomp-cone T f g c)) | ||
| ( triangle-map-postcomp-cone-standard-pullback T f g c) | ||
| ( is-equiv-map-postcomp-cone-standard-pullback f g T) | ||
| ( universal-property-pullback-is-pullback f g c is-pb-c T) | ||
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| abstract | ||
| is-pullback-is-pullback-postcomp : | ||
| {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} | ||
| (f : A → X) (g : B → X) (c : cone f g C) → | ||
| ( {l5 : Level} (T : UU l5) → | ||
| is-pullback (postcomp T f) (postcomp T g) (postcomp-cone T f g c)) → | ||
| is-pullback f g c | ||
| is-pullback-is-pullback-postcomp f g c is-pb-postcomp = | ||
| is-pullback-universal-property-pullback f g c | ||
| ( λ T → | ||
| is-equiv-left-map-triangle | ||
| ( cone-map f g c) | ||
| ( map-postcomp-cone-standard-pullback f g T) | ||
| ( gap (f ∘_) (g ∘_) (postcomp-cone T f g c)) | ||
| ( triangle-map-postcomp-cone-standard-pullback T f g c) | ||
| ( is-pb-postcomp T) | ||
| ( is-equiv-map-postcomp-cone-standard-pullback f g T)) | ||
| ``` | ||
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| ### Cones satisfying the universal property of pullbacks are closed under postcomposition exponentiation | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 l3 l4 l5 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} | ||
| (T : UU l5) (f : A → X) (g : B → X) (c : cone f g C) | ||
| where | ||
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| universal-property-pullback-postcomp : | ||
| universal-property-pullback f g c → | ||
| universal-property-pullback | ||
| ( postcomp T f) | ||
| ( postcomp T g) | ||
| ( postcomp-cone T f g c) | ||
| universal-property-pullback-postcomp H = | ||
| universal-property-pullback-is-pullback | ||
| ( postcomp T f) | ||
| ( postcomp T g) | ||
| ( postcomp-cone T f g c) | ||
| ( is-pullback-postcomp-is-pullback f g c | ||
| ( is-pullback-universal-property-pullback f g c H) | ||
| ( T)) | ||
| ``` | ||
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| ## Table of files about pullbacks | ||
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| The following table lists files that are about pullbacks as a general concept. | ||
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| {{#include tables/pullbacks.md}} |
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