Dependent cocones under spans

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-06-10.
Last modified on 2024-01-11.

module synthetic-homotopy-theory.dependent-cocones-under-spans where
Imports
open import foundation.action-on-identifications-dependent-functions
open import foundation.action-on-identifications-functions
open import foundation.contractible-types
open import foundation.dependent-identifications
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.functoriality-dependent-pair-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.homotopies
open import foundation.homotopy-induction
open import foundation.identity-types
open import foundation.structure-identity-principle
open import foundation.torsorial-type-families
open import foundation.transport-along-identifications
open import foundation.universe-levels

open import synthetic-homotopy-theory.cocones-under-spans

Idea

Consider a span 𝒮 := (A <-- S --> B) and a cocone structure c of 𝒮 into a type X. Furthermore, consider a type family P over X. In this case we may consider dependent cocone structures on P over c.

A dependent cocone d over (i , j , H) on P consists of two dependent functions

  i' : (a : A) → P (i a)
  j' : (b : B) → P (j b)

and a family of dependent identifications

  (s : S) → dependent-identification P (H s) (i' (f s)) (j' (g s)).

Definitions

Dependent cocones

module _
  { l1 l2 l3 l4 l5 : Level} {S : UU l1} {A : UU l2} {B : UU l3} {X : UU l4}
  ( f : S  A) (g : S  B) (c : cocone f g X) (P : X  UU l5)
  where

  dependent-cocone : UU (l1  l2  l3  l5)
  dependent-cocone =
    Σ ( (a : A)  P (horizontal-map-cocone f g c a))
      ( λ hA 
        Σ ( (b : B)  P (vertical-map-cocone f g c b))
          ( λ hB 
            (s : S) 
            dependent-identification P
              ( coherence-square-cocone f g c s)
              ( hA (f s))
              ( hB (g s))))

  module _
    (d : dependent-cocone)
    where

    horizontal-map-dependent-cocone :
      (a : A)  P (horizontal-map-cocone f g c a)
    horizontal-map-dependent-cocone = pr1 d

    vertical-map-dependent-cocone :
      (b : B)  P (vertical-map-cocone f g c b)
    vertical-map-dependent-cocone = pr1 (pr2 d)

    coherence-square-dependent-cocone :
      (s : S) 
      dependent-identification P
        ( coherence-square-cocone f g c s)
        ( horizontal-map-dependent-cocone (f s))
        ( vertical-map-dependent-cocone (g s))
    coherence-square-dependent-cocone = pr2 (pr2 d)

Postcomposing dependent cocones with maps

dependent-cocone-map :
  { l1 l2 l3 l4 l5 : Level} {S : UU l1} {A : UU l2} {B : UU l3} {X : UU l4}
  ( f : S  A) (g : S  B) (c : cocone f g X) (P : X  UU l5) 
  ( (x : X)  P x)  dependent-cocone f g c P
pr1 (dependent-cocone-map f g c P h) a =
  h (horizontal-map-cocone f g c a)
pr1 (pr2 (dependent-cocone-map f g c P h)) b =
  h (vertical-map-cocone f g c b)
pr2 (pr2 (dependent-cocone-map f g c P h)) s =
  apd h (coherence-square-cocone f g c s)

Properties

Characterization of identifications of dependent cocones

module _
  {l1 l2 l3 l4 l5 : Level}
  {S : UU l1} {A : UU l2} {B : UU l3} (f : S  A) (g : S  B)
  {X : UU l4} (c : cocone f g X)
  (P : X  UU l5) (d : dependent-cocone f g c P)
  where

  coherence-htpy-dependent-cocone :
    ( d' : dependent-cocone f g c P)
    ( K :
      horizontal-map-dependent-cocone f g c P d ~
      horizontal-map-dependent-cocone f g c P d')
    ( L :
      vertical-map-dependent-cocone f g c P d ~
      vertical-map-dependent-cocone f g c P d') 
    UU (l1  l5)
  coherence-htpy-dependent-cocone d' K L =
    (s : S) 
    ( ( coherence-square-dependent-cocone f g c P d s)  (L (g s))) 
    ( ( ap (tr P (coherence-square-cocone f g c s)) (K (f s))) 
      ( coherence-square-dependent-cocone f g c P d' s))

  htpy-dependent-cocone :
    (d' : dependent-cocone f g c P)  UU (l1  l2  l3  l5)
  htpy-dependent-cocone d' =
    Σ ( horizontal-map-dependent-cocone f g c P d ~
        horizontal-map-dependent-cocone f g c P d')
      ( λ K 
        Σ ( vertical-map-dependent-cocone f g c P d ~
            vertical-map-dependent-cocone f g c P d')
          ( coherence-htpy-dependent-cocone d' K))

  refl-htpy-dependent-cocone :
    htpy-dependent-cocone d
  pr1 refl-htpy-dependent-cocone = refl-htpy
  pr1 (pr2 refl-htpy-dependent-cocone) = refl-htpy
  pr2 (pr2 refl-htpy-dependent-cocone) = right-unit-htpy

  htpy-eq-dependent-cocone :
    (d' : dependent-cocone f g c P)  d  d'  htpy-dependent-cocone d'
  htpy-eq-dependent-cocone .d refl = refl-htpy-dependent-cocone

  module _
    (d' : dependent-cocone f g c P)
    (p : d  d')
    where

    horizontal-htpy-eq-dependent-cocone :
      horizontal-map-dependent-cocone f g c P d ~
      horizontal-map-dependent-cocone f g c P d'
    horizontal-htpy-eq-dependent-cocone =
      pr1 (htpy-eq-dependent-cocone d' p)

    vertical-htpy-eq-dependent-cocone :
      vertical-map-dependent-cocone f g c P d ~
      vertical-map-dependent-cocone f g c P d'
    vertical-htpy-eq-dependent-cocone =
      pr1 (pr2 (htpy-eq-dependent-cocone d' p))

    coherence-square-htpy-eq-dependent-cocone :
      coherence-htpy-dependent-cocone d'
        ( horizontal-htpy-eq-dependent-cocone)
        ( vertical-htpy-eq-dependent-cocone)
    coherence-square-htpy-eq-dependent-cocone =
      pr2 (pr2 (htpy-eq-dependent-cocone d' p))

  abstract
    is-torsorial-htpy-dependent-cocone :
      is-torsorial htpy-dependent-cocone
    is-torsorial-htpy-dependent-cocone =
      is-torsorial-Eq-structure
        ( is-torsorial-htpy (horizontal-map-dependent-cocone f g c P d))
        ( horizontal-map-dependent-cocone f g c P d , refl-htpy)
        ( is-torsorial-Eq-structure
          ( is-torsorial-htpy (vertical-map-dependent-cocone f g c P d))
          ( vertical-map-dependent-cocone f g c P d , refl-htpy)
          ( is-contr-equiv
            ( Σ ( (s : S) 
                  dependent-identification P
                    ( coherence-square-cocone f g c s)
                    ( horizontal-map-dependent-cocone f g c P d (f s))
                    ( vertical-map-dependent-cocone f g c P d (g s)))
                ( λ γ  coherence-square-dependent-cocone f g c P d ~ γ))
            ( equiv-tot (equiv-concat-htpy inv-htpy-right-unit-htpy))
            ( is-torsorial-htpy
              ( coherence-square-dependent-cocone f g c P d))))

  abstract
    is-equiv-htpy-eq-dependent-cocone :
      (d' : dependent-cocone f g c P)  is-equiv (htpy-eq-dependent-cocone d')
    is-equiv-htpy-eq-dependent-cocone =
      fundamental-theorem-id
        ( is-torsorial-htpy-dependent-cocone)
        ( htpy-eq-dependent-cocone)

    eq-htpy-dependent-cocone :
      (d' : dependent-cocone f g c P)  htpy-dependent-cocone d'  d  d'
    eq-htpy-dependent-cocone d' =
      map-inv-is-equiv (is-equiv-htpy-eq-dependent-cocone d')

    is-section-eq-htpy-dependent-cocone :
      (d' : dependent-cocone f g c P) 
      ( htpy-eq-dependent-cocone d'  eq-htpy-dependent-cocone d') ~ id
    is-section-eq-htpy-dependent-cocone d' =
      is-section-map-inv-is-equiv (is-equiv-htpy-eq-dependent-cocone d')

    is-retraction-eq-htpy-dependent-cocone :
      (d' : dependent-cocone f g c P) 
      ( eq-htpy-dependent-cocone d'  htpy-eq-dependent-cocone d') ~ id
    is-retraction-eq-htpy-dependent-cocone d' =
      is-retraction-map-inv-is-equiv (is-equiv-htpy-eq-dependent-cocone d')

Recent changes