The induction principle of pushouts

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-06-10.
Last modified on 2023-06-15.

module synthetic-homotopy-theory.induction-principle-pushouts where
Imports 
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.sections
open import foundation.universe-levels

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

Idea

The induction principle of pushouts asserts that for every dependent cocone of a type family P over a type X equipped with a cocone c there is a section of P corresponding to c. More precisely, it asserts that the map

  dependent-cocone-map f g c P : ((x : X) → P x) → dependent-cocone f g c P

has a section.

The fact that the induction principle of pushouts is logically equivalent to the dependent universal property of pushouts is shown in dependent-universal-property-pushouts.

Definition

The induction principle of pushouts

induction-principle-pushout :
  { l1 l2 l3 l4 : Level} (l : Level) →
  { S : UU l1} {A : UU l2} {B : UU l3} {X : UU l4} →
  ( f : S → A) (g : S → B) (c : cocone f g X) →
  UU (lsuc l ⊔ l1 ⊔ l2 ⊔ l3 ⊔ l4)
induction-principle-pushout l {X = X} f g c =
  (P : X → UU l) → section (dependent-cocone-map f g c P)

module _
  { l1 l2 l3 l4 l : Level} {S : UU l1} {A : UU l2} {B : UU l3} {X : UU l4}
  ( f : S → A) (g : S → B) (c : cocone f g X)
  ( ind-c : induction-principle-pushout l f g c)
  ( P : X → UU l)
  where

  ind-induction-principle-pushout : dependent-cocone f g c P → (x : X) → P x
  ind-induction-principle-pushout = pr1 (ind-c P)

  compute-ind-induction-principle-pushout :
    (h : dependent-cocone f g c P) →
    htpy-dependent-cocone f g c P
      ( dependent-cocone-map f g c P (ind-induction-principle-pushout h))
      ( h)
  compute-ind-induction-principle-pushout h =
    htpy-eq-dependent-cocone f g c P
      ( dependent-cocone-map f g c P (ind-induction-principle-pushout h))
      ( h)
      ( pr2 (ind-c P) h)

  left-compute-ind-induction-principle-pushout :
    ( h : dependent-cocone f g c P) (a : A) →
    ind-induction-principle-pushout h (horizontal-map-cocone f g c a) ＝
    horizontal-map-dependent-cocone f g c P h a
  left-compute-ind-induction-principle-pushout h =
    pr1 (compute-ind-induction-principle-pushout h)

  right-compute-ind-induction-principle-pushout :
    ( h : dependent-cocone f g c P) (b : B) →
    ind-induction-principle-pushout h (vertical-map-cocone f g c b) ＝
    vertical-map-dependent-cocone f g c P h b
  right-compute-ind-induction-principle-pushout h =
    pr1 (pr2 (compute-ind-induction-principle-pushout h))

  path-compute-ind-induction-principle-pushout :
    (h : dependent-cocone f g c P) →
    coherence-htpy-dependent-cocone f g c P
      ( dependent-cocone-map f g c P (ind-induction-principle-pushout h))
      ( h)
      ( left-compute-ind-induction-principle-pushout h)
      ( right-compute-ind-induction-principle-pushout h)
  path-compute-ind-induction-principle-pushout h =
    pr2 (pr2 (compute-ind-induction-principle-pushout h))

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