Commuting triangles of maps

module foundation.commuting-triangles-of-maps where

open import foundation-core.commuting-triangles-of-maps public

open import foundation.functoriality-dependent-function-types
open import foundation.identity-types

open import foundation-core.equivalences
open import foundation-core.universe-levels

Idea

A triangle of maps

 A ----> B
  \     /
   \   /
    V V
     X

is said to commute if there is a homotopy between the map on the left and the composite map.

Properties

Top map is an equivalence

If the top map is an equivalence, then there is an equivalence between the coherence triangle with the map of the equivalence as with the inverse map of the equivalence.

module _
  {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3}
  (left : A → X) (right : B → X) (e : A ≃ B)
  where

  equiv-coherence-triangle-maps-inv-top :
    coherence-triangle-maps left right (map-equiv e) ≃
    coherence-triangle-maps' right left (map-inv-equiv e)
  equiv-coherence-triangle-maps-inv-top =
    equiv-Π
      ( λ b → left (map-inv-equiv e b) ＝ right b)
      ( e)
      ( λ a →
        equiv-concat
          ( ap left (isretr-map-inv-equiv e a))
          ( right (map-equiv e a)))