Commuting cubes of maps

Content created by Fredrik Bakke, Egbert Rijke and Jonathan Prieto-Cubides.

Created on 2023-02-18.
Last modified on 2023-09-17.

module foundation.commuting-cubes-of-maps where
Imports 
open import foundation.action-on-identifications-functions
open import foundation.commuting-hexagons-of-identifications
open import foundation.commuting-squares-of-maps
open import foundation.cones-over-cospans
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import foundation-core.function-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.whiskering-homotopies

Idea

We specify the type of the homotopy witnessing that a cube commutes. Imagine that the cube is presented as a lattice

            *
          / | \
         /  |  \
        /   |   \
       *    *    *
       |\ /   \ /|
       | \     ‌/ |
       |/ \   / \|
       *    *    *
        \   |   /
         \  |  /
          \ | /
            *

with all maps pointing in the downwards direction. Presented in this way, a cube of maps has a top face, a back-left face, a back-right face, a front-left face, a front-right face, and a bottom face, all of which are homotopies. An element of type coherence-cube-maps is a homotopy filling the cube.

Definition

module _
  {l1 l2 l3 l4 l1' l2' l3' l4' : Level}
  {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4}
  (f : A → B) (g : A → C) (h : B → D) (k : C → D)
  {A' : UU l1'} {B' : UU l2'} {C' : UU l3'} {D' : UU l4'}
  (f' : A' → B') (g' : A' → C') (h' : B' → D') (k' : C' → D')
  (hA : A' → A) (hB : B' → B) (hC : C' → C) (hD : D' → D)
  where

  coherence-cube-maps :
    (top : (h' ∘ f') ~ (k' ∘ g'))
    (back-left : (f ∘ hA) ~ (hB ∘ f'))
    (back-right : (g ∘ hA) ~ (hC ∘ g'))
    (front-left : (h ∘ hB) ~ (hD ∘ h'))
    (front-right : (k ∘ hC) ~ (hD ∘ k'))
    (bottom : (h ∘ f) ~ (k ∘ g)) →
    UU (l4 ⊔ l1')
  coherence-cube-maps top back-left back-right front-left front-right bottom =
    (a' : A') →
    coherence-hexagon
      ( ap h (back-left a'))
      ( front-left (f' a'))
      ( ap hD (top a'))
      ( bottom (hA a'))
      ( ap k (back-right a'))
      ( front-right (g' a'))

Symmetries of commuting cubes

The symmetry group D₃ acts on a cube. However, the coherence filling a cube needs to be modified to show that the rotated/reflected cube again commutes. In the following definitions we provide the homotopies witnessing that the rotated/reflected cubes again commute.

Note: although in principle it ought to be enough to show this for the generators of the symmetry group D₃, in practice it is more straightforward to just do the work for each of the symmetries separately. One reason is that some of the homotopies witnessing that the faces commute will be inverted as the result of an application of a symmetry. Inverting a homotopy twice results in a new homotopy that is only homotopic to the original homotopy.

module _
  {l1 l2 l3 l4 l1' l2' l3' l4' : Level}
  {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4}
  (f : A → B) (g : A → C) (h : B → D) (k : C → D)
  {A' : UU l1'} {B' : UU l2'} {C' : UU l3'} {D' : UU l4'}
  (f' : A' → B') (g' : A' → C') (h' : B' → D') (k' : C' → D')
  (hA : A' → A) (hB : B' → B) (hC : C' → C) (hD : D' → D)
  (top : coherence-square-maps g' f' k' h')
  (back-left : coherence-square-maps f' hA hB f)
  (back-right : coherence-square-maps g' hA hC g)
  (front-left : coherence-square-maps h' hB hD h)
  (front-right : coherence-square-maps k' hC hD k)
  (bottom : coherence-square-maps g f k h)
  (c :
    coherence-cube-maps
      f g h k f' g' h' k' hA hB hC hD
      top back-left back-right front-left front-right bottom)
  where

  coherence-cube-maps-rotate-120 :
    coherence-cube-maps hC k' k hD hA f' f hB g' g h' h
      ( back-left)
      ( inv-htpy back-right)
      ( inv-htpy top)
      ( inv-htpy bottom)
      ( inv-htpy front-left)
      ( front-right)
  coherence-cube-maps-rotate-120 a' =
    ( ap (λ t → t ∙ (ap h (back-left a')))
      ( ap (λ t' → t' ∙ inv (bottom (hA a')))
        ( ap-inv k (back-right a')))) ∙
    ( ( hexagon-rotate-120
        ( ap h (back-left a'))
        ( front-left (f' a'))
        ( ap hD (top a'))
        ( bottom (hA a'))
        ( ap k (back-right a'))
        ( front-right (g' a'))
        ( c a')) ∙
      ( inv
        ( ap (λ t → (front-right (g' a')) ∙ t)
          ( ap (λ t' → t' ∙ inv (front-left (f' a')))
            ( ap-inv hD (top a'))))))

  coherence-cube-maps-rotate-240 :
    coherence-cube-maps h' hB hD h g' hA hC g f' k' f k
      ( inv-htpy back-right)
      ( top)
      ( inv-htpy back-left)
      ( inv-htpy front-right)
      ( bottom)
      ( inv-htpy front-left)
  coherence-cube-maps-rotate-240 a' =
    ( ap (λ t → _ ∙ t) (ap-inv k (back-right a'))) ∙
    ( ( hexagon-rotate-240
        ( ap h (back-left a'))
        ( front-left (f' a'))
        ( ap hD (top a'))
        ( bottom (hA a'))
        ( ap k (back-right a'))
        ( front-right (g' a'))
        ( c a')) ∙
      ( inv
        ( ap
          ( λ t → inv (front-left (f' a')) ∙ t)
          ( ap (λ t' → t' ∙ _) (ap-inv h (back-left a'))))))

  coherence-cube-maps-mirror-A :
    coherence-cube-maps g f k h g' f' k' h' hA hC hB hD
      ( inv-htpy top)
      ( back-right)
      ( back-left)
      ( front-right)
      ( front-left)
      ( inv-htpy bottom)
  coherence-cube-maps-mirror-A a' =
    ( ap (λ t → _ ∙ t) (ap-inv hD (top a'))) ∙
    ( hexagon-mirror-A
      ( ap h (back-left a'))
      ( front-left (f' a'))
      ( ap hD (top a'))
      ( bottom (hA a'))
      ( ap k (back-right a'))
      ( front-right (g' a'))
      ( c a'))

  coherence-cube-maps-mirror-B :
    coherence-cube-maps hB h' h hD hA g' g hC f' f k' k
      ( back-right)
      ( inv-htpy back-left)
      ( top)
      ( bottom)
      ( inv-htpy front-right)
      ( front-left)
  coherence-cube-maps-mirror-B a' =
    ( ap (λ t → t ∙ (ap k (back-right a')))
      ( ap (λ t → t ∙ _) (ap-inv h (back-left a')))) ∙
    ( hexagon-mirror-B
      ( ap h (back-left a'))
      ( front-left (f' a'))
      ( ap hD (top a'))
      ( bottom (hA a'))
      ( ap k (back-right a'))
      ( front-right (g' a'))
      ( c a'))

  coherence-cube-maps-mirror-C :
    coherence-cube-maps k' hC hD k f' hA hB f g' h' g h
      ( inv-htpy back-left)
      ( inv-htpy top)
      ( inv-htpy back-right)
      ( inv-htpy front-left)
      ( inv-htpy bottom)
      ( inv-htpy front-right)
  coherence-cube-maps-mirror-C a' =
    ( ap
      ( λ t → (t ∙ inv (front-left (f' a'))) ∙ (ap h (inv (back-left a'))))
      ( ap-inv hD (top a'))) ∙
    ( ( ap (λ t → _ ∙ t) (ap-inv h (back-left a'))) ∙
      ( ( hexagon-mirror-C
          ( ap h (back-left a'))
          ( front-left (f' a'))
          ( ap hD (top a'))
          ( bottom (hA a'))
          ( ap k (back-right a'))
          ( front-right (g' a'))
          ( c a')) ∙
        ( inv
          ( ap
            ( λ t → inv (front-right (g' a')) ∙ t)
            ( ap (λ t' → t' ∙ _) (ap-inv k (back-right a')))))))

Rectangles in commuting cubes

module _
  {l1 l2 l3 l4 l1' l2' l3' l4' : Level}
  {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4}
  (f : A → B) (g : A → C) (h : B → D) (k : C → D)
  {A' : UU l1'} {B' : UU l2'} {C' : UU l3'} {D' : UU l4'}
  (f' : A' → B') (g' : A' → C') (h' : B' → D') (k' : C' → D')
  (hA : A' → A) (hB : B' → B) (hC : C' → C) (hD : D' → D)
  (top : coherence-square-maps g' f' k' h')
  (back-left : coherence-square-maps f' hA hB f)
  (back-right : coherence-square-maps g' hA hC g)
  (front-left : coherence-square-maps h' hB hD h)
  (front-right : coherence-square-maps k' hC hD k)
  (bottom : coherence-square-maps g f k h)
  where

  rectangle-left-cube :
    ((h ∘ f) ∘ hA) ~ (hD ∘ (h' ∘ f'))
  rectangle-left-cube =
    pasting-horizontal-coherence-square-maps
      f' h' hA hB hD f h back-left front-left

  rectangle-right-cube :
    ((k ∘ g) ∘ hA) ~ (hD ∘ (k' ∘ g'))
  rectangle-right-cube =
    pasting-horizontal-coherence-square-maps
      g' k' hA hC hD g k back-right front-right

  coherence-htpy-parallel-cone-rectangle-left-rectangle-right-cube :
    (c : coherence-cube-maps f g h k f' g' h' k' hA hB hC hD
      top back-left back-right front-left front-right bottom) →
    coherence-htpy-parallel-cone
      ( bottom)
      ( refl-htpy' hD)
      ( hA , h' ∘ f' , rectangle-left-cube)
      ( hA , k' ∘ g' , rectangle-right-cube)
      ( refl-htpy' hA)
      ( top)
  coherence-htpy-parallel-cone-rectangle-left-rectangle-right-cube c =
    ( λ a' →
      ( ap
        ( concat
          ( rectangle-left-cube a')
          ( hD (k' (g' a'))))
        ( right-unit))) ∙h
    ( c)

  rectangle-top-front-left-cube :
    ((h ∘ hB) ∘ f') ~ ((hD ∘ k') ∘ g')
  rectangle-top-front-left-cube =
    ( front-left ·r f') ∙h (hD ·l top)

  rectangle-back-right-bottom-cube :
    ((h ∘ f) ∘ hA) ~ ((k ∘ hC) ∘ g')
  rectangle-back-right-bottom-cube =
    ( bottom ·r hA) ∙h (k ·l back-right)

  rectangle-top-front-right-cube :
    ((hD ∘ h') ∘ f') ~ ((k ∘ hC) ∘ g')
  rectangle-top-front-right-cube =
    (hD ·l top) ∙h (inv-htpy (front-right) ·r g')

  rectangle-back-left-bottom-cube :
    ((h ∘ hB) ∘ f') ~ ((k ∘ g) ∘ hA)
  rectangle-back-left-bottom-cube =
    (h ·l (inv-htpy back-left)) ∙h (bottom ·r hA)

In analogy to the coherence coherence-htpy-parallel-cone-rectangle-left-rectangle-right-cube we also expect to be able to construct a coherence

  coherence-htpy-parallel-cone-rectangle-top-fl-rectangle-br-bot-cube :
    (c : coherence-cube-maps f g h k f' g' h' k' hA hB hC hD
      top back-left back-right front-left front-right bottom) →
    coherence-htpy-parallel-cone
      ( inv-htpy front-right)
      ( refl-htpy' h)
      ( g' , hB ∘ f' , inv-htpy (rectangle-top-front-left-cube))
      ( g' , f ∘ hA , inv-htpy (rectangle-back-right-bottom-cube))
      ( refl-htpy' g')
      ( inv-htpy back-left)

Any coherence of commuting cubes induces a coherence of parallel cones

module _
  {l1 l2 l3 l4 l1' l2' l3' l4' : Level}
  {A : UU l1} {B : UU l2} {C : UU l3} {D : UU l4}
  (f : A → B) (g : A → C) (h : B → D) (k : C → D)
  {A' : UU l1'} {B' : UU l2'} {C' : UU l3'} {D' : UU l4'}
  (f' : A' → B') (g' : A' → C') (h' : B' → D') (k' : C' → D')
  (hA : A' → A) (hB : B' → B) (hC : C' → C) (hD : D' → D)
  (top : coherence-square-maps g' f' k' h')
  (back-left : coherence-square-maps f' hA hB f)
  (back-right : coherence-square-maps g' hA hC g)
  (front-left : coherence-square-maps h' hB hD h)
  (front-right : coherence-square-maps k' hC hD k)
  (bottom : coherence-square-maps g f k h)
  where

  coherence-htpy-parallel-cone-coherence-cube-maps :
    ( c :
      coherence-cube-maps
        f g h k f' g' h' k' hA hB hC hD
        top back-left back-right front-left front-right bottom) →
    coherence-htpy-parallel-cone
      ( front-left)
      ( refl-htpy' k)
      ( ( f') ,
        ( g ∘ hA) ,
        ( rectangle-back-left-bottom-cube
          f g h k f' g' h' k' hA hB hC hD
          top back-left back-right front-left front-right bottom))
      ( ( f') ,
        ( hC ∘ g') ,
        ( rectangle-top-front-right-cube
          f g h k f' g' h' k' hA hB hC hD
          top back-left back-right front-left front-right bottom))
      ( refl-htpy' f')
      ( back-right)
  coherence-htpy-parallel-cone-coherence-cube-maps c =
    ( assoc-htpy
      ( h ·l (inv-htpy back-left))
      ( bottom ·r hA)
      ( (k ·l back-right) ∙h (refl-htpy' (k ∘ (hC ∘ g'))))) ∙h
    ( ( ap-concat-htpy'
        ( h ·l (inv-htpy back-left))
        ( inv-htpy (h ·l back-left))
        ( _)
        ( left-whisk-inv-htpy h back-left)) ∙h
      ( inv-htpy-left-transpose-htpy-concat (h ·l back-left) _ _
        ( ( (inv-htpy-assoc-htpy (h ·l back-left) (front-left ·r f') _) ∙h
            ( ( inv-htpy-assoc-htpy
                ( (h ·l back-left) ∙h (front-left ·r f'))
                ( hD ·l top)
                ( (inv-htpy front-right) ·r g')) ∙h
              ( inv-htpy-right-transpose-htpy-concat _ (front-right ·r g') _
                ( (assoc-htpy (bottom ·r hA) _ _) ∙h (inv-htpy c))))) ∙h
          ( ap-concat-htpy (bottom ·r hA) _ _ inv-htpy-right-unit-htpy))))

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