Descent data for type families of identity 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-identity-types-over-pushouts where
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.identity-types
open import foundation.span-diagrams
open import foundation.transport-along-identifications
open import foundation.universe-levels

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


Given a cocone under a span diagram

    S -----> B
    |        |
  f |        | j
    ∨        ∨
    A -----> X

and a point x₀ : X, the type family of identity types based at x₀, x ↦ (x₀ = x), is characterized by the descent data (IA, IB, IS), where IA and IB are families of identity types

  IA a := (x₀ = ia)
  IB b := (x₀ = jb),

and the gluing data IS s : (x₀ = ifs) ≃ (x₀ = jgs) is given by concatenation with the coherence of the cocone H s : ifs = jgs.


module _
  {l1 l2 l3 l4 : Level} {𝒮 : span-diagram l1 l2 l3}
  {X : UU l4} (c : cocone-span-diagram 𝒮 X)
  (x₀ : X)

  family-cocone-identity-type-pushout : X  UU l4
  family-cocone-identity-type-pushout x = x₀  x

  descent-data-identity-type-pushout : descent-data-pushout 𝒮 l4
  pr1 descent-data-identity-type-pushout a =
    x₀  horizontal-map-cocone _ _ c a
  pr1 (pr2 descent-data-identity-type-pushout) b =
    x₀  vertical-map-cocone _ _ c b
  pr2 (pr2 descent-data-identity-type-pushout) s =
    equiv-concat' x₀ (coherence-square-cocone _ _ c s)

  equiv-descent-data-identity-type-pushout :
      ( descent-data-family-cocone-span-diagram c
        ( family-cocone-identity-type-pushout))
      ( descent-data-identity-type-pushout)
  pr1 equiv-descent-data-identity-type-pushout a = id-equiv
  pr1 (pr2 equiv-descent-data-identity-type-pushout) b = id-equiv
  pr2 (pr2 equiv-descent-data-identity-type-pushout) s =
    tr-Id-right (coherence-square-cocone _ _ c s)

  family-with-descent-data-identity-type-pushout :
    family-with-descent-data-pushout c l4
  pr1 family-with-descent-data-identity-type-pushout =
  pr1 (pr2 family-with-descent-data-identity-type-pushout) =
  pr2 (pr2 family-with-descent-data-identity-type-pushout) =

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