Descent data for type families of function types over pushouts
Content created by Vojtěch Štěpančík.
Created on 2024-06-05.
Last modified on 2024-06-05.
{-# OPTIONS --lossy-unification #-} module synthetic-homotopy-theory.descent-data-function-types-over-pushouts where
Imports
open import foundation.commuting-squares-of-maps open import foundation.commuting-triangles-of-maps open import foundation.contractible-maps open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps open import foundation.function-extensionality open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.homotopy-algebra open import foundation.postcomposition-functions open import foundation.span-diagrams open import foundation.transport-along-identifications open import foundation.universal-property-equivalences open import foundation.universe-levels open import foundation.whiskering-homotopies-composition open import synthetic-homotopy-theory.cocones-under-spans open import synthetic-homotopy-theory.descent-data-pushouts open import synthetic-homotopy-theory.equivalences-descent-data-pushouts open import synthetic-homotopy-theory.families-descent-data-pushouts open import synthetic-homotopy-theory.morphisms-descent-data-pushouts open import synthetic-homotopy-theory.sections-descent-data-pushouts open import synthetic-homotopy-theory.universal-property-pushouts
Idea
Given two
families with descent data
for pushouts P ≈ (PA, PB, PS) and
R ≈ (RA, RB, RS), we show that
fiberwise maps (x : X) → P x → R x
correspond to
morphisms
(PA, PB, PS) → (RA, RB, RS).
Proof: We first characterize the type family x ↦ (P x → R x). The
corresponding descent data
consists of the type families
a ↦ (PA a → RA a)
b ↦ (PB b → RB b),
and the gluing data sending h : PA (fs) → RA (fs) to
(RS s ∘ h ∘ (PS s)⁻¹) : PB (gs) → RB (gs).
Note: It is important to differentiate between families of function types,
i.e. a type family that to every x : X assigns the type P x → R x, and
families of functions, i.e. a family that to every x : X assigns a
function from P x to R x`. Descent data plays the role of a family of types,
so it makes sense to talk about “descent data corresponding to a family of
function types”, but it doesn’t make sense to talk about “descent data
corresponding to a family of functions”. The kind of data that corresponds to
families of functions are the sections of the descent data of a family of
function types.
It suffices to show that the sections correspond to morphisms. The first two components of a section and a morphism agree, namely they are
sA : (a : A) → PA a → RA a
sB : (b : B) → PB b → RB b,
respectively. The coherence datum of a section has the type
(s : S) → RS s ∘ sA (fs) ∘ (RS s)⁻¹ = sB (gs),
which we can massage into a coherence of the morphism as
RS s ∘ sA (fs) ∘ (PS s)⁻¹ = sB (gs)
≃ RS s ∘ sA (fs) ∘ (PS s)⁻¹ ~ sB (gs) by function extensionality
≃ RS s ∘ sA (fs) ~ sB (gs) ∘ PS s by transposition of (PS s)
≃ sB (gs) ∘ PS s ~ RS s ∘ sA (fs) by symmetry of homotopies.
Definitions
The type family of fiberwise functions with corresponding descent data for pushouts
module _ {l1 l2 l3 l4 l5 l6 : Level} {𝒮 : span-diagram l1 l2 l3} {X : UU l4} {c : cocone-span-diagram 𝒮 X} (P : family-with-descent-data-pushout c l5) (R : family-with-descent-data-pushout c l6) where family-cocone-function-type-pushout : X → UU (l5 ⊔ l6) family-cocone-function-type-pushout x = family-cocone-family-with-descent-data-pushout P x → family-cocone-family-with-descent-data-pushout R x descent-data-function-type-pushout : descent-data-pushout 𝒮 (l5 ⊔ l6) pr1 descent-data-function-type-pushout a = left-family-family-with-descent-data-pushout P a → left-family-family-with-descent-data-pushout R a pr1 (pr2 descent-data-function-type-pushout) b = right-family-family-with-descent-data-pushout P b → right-family-family-with-descent-data-pushout R b pr2 (pr2 descent-data-function-type-pushout) s = ( equiv-postcomp _ ( equiv-family-family-with-descent-data-pushout R s)) ∘e ( equiv-precomp ( inv-equiv (equiv-family-family-with-descent-data-pushout P s)) ( _)) left-equiv-equiv-descent-data-function-type-pushout : (a : domain-span-diagram 𝒮) → ( family-cocone-family-with-descent-data-pushout P ( horizontal-map-cocone _ _ c a) → family-cocone-family-with-descent-data-pushout R ( horizontal-map-cocone _ _ c a)) ≃ ( left-family-family-with-descent-data-pushout P a → left-family-family-with-descent-data-pushout R a) left-equiv-equiv-descent-data-function-type-pushout a = ( equiv-postcomp _ ( left-equiv-family-with-descent-data-pushout R a)) ∘e ( equiv-precomp ( inv-equiv (left-equiv-family-with-descent-data-pushout P a)) ( _)) right-equiv-equiv-descent-data-function-type-pushout : (b : codomain-span-diagram 𝒮) → ( family-cocone-family-with-descent-data-pushout P ( vertical-map-cocone _ _ c b) → family-cocone-family-with-descent-data-pushout R ( vertical-map-cocone _ _ c b)) ≃ ( right-family-family-with-descent-data-pushout P b → right-family-family-with-descent-data-pushout R b) right-equiv-equiv-descent-data-function-type-pushout b = ( equiv-postcomp _ ( right-equiv-family-with-descent-data-pushout R b)) ∘e ( equiv-precomp ( inv-equiv (right-equiv-family-with-descent-data-pushout P b)) ( _)) coherence-equiv-descent-data-function-type-pushout : (s : spanning-type-span-diagram 𝒮) → coherence-square-maps ( map-equiv ( left-equiv-equiv-descent-data-function-type-pushout ( left-map-span-diagram 𝒮 s))) ( tr ( family-cocone-function-type-pushout) ( coherence-square-cocone _ _ c s)) ( map-family-descent-data-pushout ( descent-data-function-type-pushout) ( s)) ( map-equiv ( right-equiv-equiv-descent-data-function-type-pushout ( right-map-span-diagram 𝒮 s))) coherence-equiv-descent-data-function-type-pushout s = ( ( map-equiv ( right-equiv-equiv-descent-data-function-type-pushout ( right-map-span-diagram 𝒮 s))) ·l ( tr-function-type ( family-cocone-family-with-descent-data-pushout P) ( family-cocone-family-with-descent-data-pushout R) ( coherence-square-cocone _ _ c s))) ∙h ( λ h → eq-htpy ( horizontal-concat-htpy ( coherence-family-with-descent-data-pushout R s ·r h) ( coherence-square-maps-inv-equiv ( equiv-tr ( family-cocone-family-with-descent-data-pushout P) ( coherence-square-cocone _ _ c s)) ( left-equiv-family-with-descent-data-pushout P ( left-map-span-diagram 𝒮 s)) ( right-equiv-family-with-descent-data-pushout P ( right-map-span-diagram 𝒮 s)) ( equiv-family-family-with-descent-data-pushout P s) ( inv-htpy (coherence-family-with-descent-data-pushout P s))))) equiv-descent-data-function-type-pushout : equiv-descent-data-pushout ( descent-data-family-cocone-span-diagram c ( family-cocone-function-type-pushout)) ( descent-data-function-type-pushout) pr1 equiv-descent-data-function-type-pushout = left-equiv-equiv-descent-data-function-type-pushout pr1 (pr2 equiv-descent-data-function-type-pushout) = right-equiv-equiv-descent-data-function-type-pushout pr2 (pr2 equiv-descent-data-function-type-pushout) = coherence-equiv-descent-data-function-type-pushout family-with-descent-data-function-type-pushout : family-with-descent-data-pushout c (l5 ⊔ l6) pr1 family-with-descent-data-function-type-pushout = family-cocone-function-type-pushout pr1 (pr2 family-with-descent-data-function-type-pushout) = descent-data-function-type-pushout pr2 (pr2 family-with-descent-data-function-type-pushout) = equiv-descent-data-function-type-pushout
Properties
Sections of descent data for families of functions correspond to morphisms of descent data
module _ {l1 l2 l3 l4 l5 l6 : Level} {𝒮 : span-diagram l1 l2 l3} {X : UU l4} {c : cocone-span-diagram 𝒮 X} (P : family-with-descent-data-pushout c l5) (R : family-with-descent-data-pushout c l6) where hom-section-descent-data-function-type-pushout : section-descent-data-pushout (descent-data-function-type-pushout P R) → hom-descent-data-pushout ( descent-data-family-with-descent-data-pushout P) ( descent-data-family-with-descent-data-pushout R) hom-section-descent-data-function-type-pushout = tot ( λ tA → tot ( λ tB tS s → inv-htpy ( map-inv-equiv ( equiv-coherence-triangle-maps-inv-top' ( ( map-family-family-with-descent-data-pushout R s) ∘ ( tA (left-map-span-diagram 𝒮 s))) ( tB (right-map-span-diagram 𝒮 s)) ( equiv-family-family-with-descent-data-pushout P s)) ( htpy-eq (tS s))))) abstract is-equiv-hom-section-descent-data-function-type-pushout : is-equiv ( hom-section-descent-data-function-type-pushout) is-equiv-hom-section-descent-data-function-type-pushout = is-equiv-tot-is-fiberwise-equiv ( λ tA → is-equiv-tot-is-fiberwise-equiv ( λ tB → is-equiv-map-Π-is-fiberwise-equiv ( λ s → is-equiv-comp _ _ ( funext _ _) ( is-equiv-comp _ _ ( is-equiv-map-inv-equiv ( equiv-coherence-triangle-maps-inv-top' ( ( map-family-family-with-descent-data-pushout R s) ∘ ( tA (left-map-span-diagram 𝒮 s))) ( tB (right-map-span-diagram 𝒮 s)) ( equiv-family-family-with-descent-data-pushout P s))) ( is-equiv-inv-htpy _ _))))) hom-descent-data-map-family-cocone-span-diagram : ( (x : X) → family-cocone-family-with-descent-data-pushout P x → family-cocone-family-with-descent-data-pushout R x) → hom-descent-data-pushout ( descent-data-family-with-descent-data-pushout P) ( descent-data-family-with-descent-data-pushout R) hom-descent-data-map-family-cocone-span-diagram = ( hom-section-descent-data-function-type-pushout) ∘ ( section-descent-data-section-family-cocone-span-diagram ( family-with-descent-data-function-type-pushout P R)) abstract is-equiv-hom-descent-data-map-family-cocone-span-diagram : universal-property-pushout _ _ c → is-equiv hom-descent-data-map-family-cocone-span-diagram is-equiv-hom-descent-data-map-family-cocone-span-diagram up-c = is-equiv-comp _ _ ( is-equiv-section-descent-data-section-family-cocone-span-diagram _ ( up-c)) ( is-equiv-hom-section-descent-data-function-type-pushout)
As a corollary, given a morphism (hA, hB, hS) : (PA, PB, PS) → (RA, RB, RS),
there is a unique family of maps h : (x : X) → P x → R x such that its induced
morphism is homotopic to (hA, hB, hS). The homotopy provides us in particular
with the component-wise homotopies
hA a hB a
PA a --------> RA a PB b --------> RB b
| ∧ | ∧
(eᴾA a)⁻¹ | | eᴿA a (eᴾB b)⁻¹ | | eᴿB b
| | | |
∨ | ∨ |
P (ia) ------> R (ia) P (jb) ------> R (jb)
h (ia) h (jb)
which we can turn into the computation rules
eᴾA a eᴾB a
P (ia) -------> PA a P (jb) -------> PB b
| | | |
h (ia) | | hA a h (jb) | | hB b
| | | |
∨ ∨ ∨ ∨
R (ia) -------> RA a R (jb) -------> RB b
eᴿA a eᴿB b
by inverting the inverted equivalences.
module _ {l1 l2 l3 l4 l5 l6 : Level} {𝒮 : span-diagram l1 l2 l3} {X : UU l4} {c : cocone-span-diagram 𝒮 X} (up-c : universal-property-pushout _ _ c) (P : family-with-descent-data-pushout c l5) (R : family-with-descent-data-pushout c l6) (m : hom-descent-data-pushout ( descent-data-family-with-descent-data-pushout P) ( descent-data-family-with-descent-data-pushout R)) where abstract uniqueness-map-hom-descent-data-pushout : is-contr ( Σ ( (x : X) → family-cocone-family-with-descent-data-pushout P x → family-cocone-family-with-descent-data-pushout R x) ( λ h → htpy-hom-descent-data-pushout ( descent-data-family-with-descent-data-pushout P) ( descent-data-family-with-descent-data-pushout R) ( hom-descent-data-map-family-cocone-span-diagram P R h) ( m))) uniqueness-map-hom-descent-data-pushout = is-contr-equiv' ( fiber (hom-descent-data-map-family-cocone-span-diagram P R) m) ( equiv-tot ( λ h → extensionality-hom-descent-data-pushout _ _ _ m)) ( is-contr-map-is-equiv ( is-equiv-hom-descent-data-map-family-cocone-span-diagram P R up-c) ( m)) map-hom-descent-data-pushout : (x : X) → family-cocone-family-with-descent-data-pushout P x → family-cocone-family-with-descent-data-pushout R x map-hom-descent-data-pushout = pr1 (center uniqueness-map-hom-descent-data-pushout) compute-left-map-map-hom-descent-data-pushout : (a : domain-span-diagram 𝒮) → coherence-square-maps ( left-map-family-with-descent-data-pushout P a) ( map-hom-descent-data-pushout (horizontal-map-cocone _ _ c a)) ( left-map-hom-descent-data-pushout ( descent-data-family-with-descent-data-pushout P) ( descent-data-family-with-descent-data-pushout R) ( m) ( a)) ( left-map-family-with-descent-data-pushout R a) compute-left-map-map-hom-descent-data-pushout a = map-inv-equiv ( equiv-coherence-triangle-maps-inv-top' ( left-map-family-with-descent-data-pushout R a ∘ map-hom-descent-data-pushout (horizontal-map-cocone _ _ c a)) ( left-map-hom-descent-data-pushout ( descent-data-family-with-descent-data-pushout P) ( descent-data-family-with-descent-data-pushout R) ( m) ( a)) ( left-equiv-family-with-descent-data-pushout P a)) ( pr1 (pr2 (center uniqueness-map-hom-descent-data-pushout)) a) compute-right-map-map-hom-descent-data-pushout : (b : codomain-span-diagram 𝒮) → coherence-square-maps ( right-map-family-with-descent-data-pushout P b) ( map-hom-descent-data-pushout (vertical-map-cocone _ _ c b)) ( right-map-hom-descent-data-pushout ( descent-data-family-with-descent-data-pushout P) ( descent-data-family-with-descent-data-pushout R) ( m) ( b)) ( right-map-family-with-descent-data-pushout R b) compute-right-map-map-hom-descent-data-pushout b = map-inv-equiv ( equiv-coherence-triangle-maps-inv-top' ( right-map-family-with-descent-data-pushout R b ∘ map-hom-descent-data-pushout (vertical-map-cocone _ _ c b)) ( right-map-hom-descent-data-pushout ( descent-data-family-with-descent-data-pushout P) ( descent-data-family-with-descent-data-pushout R) ( m) ( b)) ( right-equiv-family-with-descent-data-pushout P b)) ( pr1 (pr2 (pr2 (center uniqueness-map-hom-descent-data-pushout))) b)
Recent changes
- 2024-06-05. Vojtěch Štěpančík. Characterization of various families over pushouts (#1148).