Commuting tetrahedra of maps

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2024-01-11.
Last modified on 2024-02-06.

module foundation.commuting-tetrahedra-of-maps where
Imports
open import foundation.universe-levels
open import foundation.whiskering-homotopies-composition

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

Idea

A commuting tetrahedron of maps is a commuting diagram of the form

  A ----------> B
  |  \       ^  |
  |    \   /    |
  |      /      |
  |    /   \    |
  V  /       v  V
  X ----------> Y.

Definition

module _
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4}
  (top : A  B) (left : A  X) (right : B  Y) (bottom : X  Y)
  (diagonal-up : X  B) (diagonal-down : A  Y)
  (upper-left : coherence-triangle-maps top diagonal-up left)
  (lower-right : coherence-triangle-maps bottom right diagonal-up)
  (upper-right : coherence-triangle-maps diagonal-down right top)
  (lower-left : coherence-triangle-maps diagonal-down bottom left)
  where

  coherence-tetrahedron-maps : UU (l1  l4)
  coherence-tetrahedron-maps =
    ( upper-right ∙h (right ·l upper-left)) ~
    ( lower-left ∙h (lower-right ·r left))

  coherence-tetrahedron-maps' : UU (l1  l4)
  coherence-tetrahedron-maps' =
    ( lower-left ∙h (lower-right ·r left)) ~
    ( upper-right ∙h (right ·l upper-left))

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