This is an archived version pinned as of the submission of my master's thesis. An up-to-date version may be found online.

Commuting prisms of maps

module foundation.commuting-prisms-of-maps where

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

open import foundation.action-on-identifications-functions
open import foundation.commuting-squares-of-maps
open import foundation.commuting-triangles-of-maps
open import foundation.composition-algebra
open import foundation.function-extensionality
open import foundation.identity-types
open import foundation.postcomposition-functions
open import foundation.precomposition-functions
open import foundation.universe-levels
open import foundation.whiskering-homotopies-composition

open import foundation-core.commuting-squares-of-homotopies
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-function-types
open import foundation-core.homotopies

Definitions

Vertical pasting of vertical prisms of maps

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

  pasting-vertical-coherence-prism-maps :
    vertical-coherence-prism-maps f g h f' g' h' hA hB hC
      ( top)
      ( front-top)
      ( right-top)
      ( left-top)
      ( mid) →
    vertical-coherence-prism-maps f' g' h' f'' g'' h'' hA' hB' hC'
      ( mid)
      ( front-bottom)
      ( right-bottom)
      ( left-bottom)
      ( bottom) →
    vertical-coherence-prism-maps f g h f'' g'' h''
      ( hA' ∘ hA)
      ( hB' ∘ hB)
      ( hC' ∘ hC)
      ( top)
      ( pasting-vertical-coherence-square-maps f hA hC f' hA' hC' f''
        ( front-top)
        ( front-bottom))
      ( pasting-vertical-coherence-square-maps g hB hC g' hB' hC' g''
        ( right-top)
        ( right-bottom))
      ( pasting-vertical-coherence-square-maps h hA hB h' hA' hB' h''
        ( left-top)
        ( left-bottom))
      ( bottom)
  pasting-vertical-coherence-prism-maps prism-top prism-bottom =
    ( ap-concat-htpy
      ( bottom ·r hA' ·r hA)
      ( commutative-pasting-vertical-pasting-horizontal-coherence-square-maps
        h g hA hB hC h' g' hA' hB' hC' h'' g''
        ( left-top)
        ( right-top)
        ( left-bottom)
        ( right-bottom))) ∙h
    ( right-whisker-concat-coherence-square-homotopies
      ( front-bottom ·r hA)
      ( bottom ·r hA' ·r hA)
      ( hC' ·l mid ·r hA)
      ( ( pasting-horizontal-coherence-square-maps
            h' g' hA' hB' hC' h'' g'' left-bottom right-bottom) ·r
        ( hA))
      ( prism-bottom ·r hA)
      ( hC' ·l ((g' ·l left-top) ∙h (right-top ·r h)))
      ) ∙h
    ( ap-concat-htpy
      ( front-bottom ·r hA)
      ( ( map-coherence-square-homotopies hC'
          ( front-top)
          ( mid ·r hA)
          (hC ·l top)
          ( pasting-horizontal-coherence-square-maps h g hA hB hC h' g'
            ( left-top)
            ( right-top))
          ( prism-top)) ∙h
        ( ap-concat-htpy
          ( hC' ·l front-top)
          ( preserves-comp-left-whisker-comp hC' hC top)))) ∙h
    ( inv-htpy-assoc-htpy
      ( front-bottom ·r hA)
      ( hC' ·l front-top)
      ( ( hC' ∘ hC) ·l top))

Properties

The two definitions of vertical prisms are equivalent

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

  equiv-rotate-vertical-coherence-prism-maps :
    vertical-coherence-prism-maps' f g h f' g' h' hA hB hC
      ( top)
      ( inv-front)
      ( inv-right)
      ( left)
      ( bottom) ≃
    vertical-coherence-prism-maps f g h f' g' h' hA hB hC
      ( top)
      ( inv-htpy inv-front)
      ( inv-htpy inv-right)
      ( left)
      ( bottom)
  equiv-rotate-vertical-coherence-prism-maps =
    equiv-Π-equiv-family
      ( λ a →
        ( equiv-concat-assoc
          ( bottom (hA a))
          ( ap g' (left a))
          ( inv (inv-right (h a))) _) ∘e
        ( equiv-right-transpose-eq-concat' _
          ( inv (inv-front a) ∙ ap hC (top a))
          ( inv-right (h a))) ∘e
        ( inv-equiv
          ( equiv-concat-assoc' _
            ( inv (inv-front a))
            ( ap hC (top a))
            ( inv-right (h a)))) ∘e
        ( equiv-left-transpose-eq-concat
          ( inv-front a)
          ( bottom (hA a) ∙ ap g' (left a))
          ( _)))

  rotate-vertical-coherence-prism-maps :
    vertical-coherence-prism-maps' f g h f' g' h' hA hB hC
      ( top)
      ( inv-front)
      ( inv-right)
      ( left)
      ( bottom) →
    vertical-coherence-prism-maps f g h f' g' h' hA hB hC
      ( top)
      ( inv-htpy inv-front)
      ( inv-htpy inv-right)
      ( left)
      ( bottom)
  rotate-vertical-coherence-prism-maps =
    map-equiv equiv-rotate-vertical-coherence-prism-maps

Commuting prisms of maps induce commuting prisms of precomposition maps

We prove this for a few different formulations of commuting prisms of maps.

The basic set-up is that, given a commuting prism of maps

          B
      h  ∧| \  g
       /  |   \
     /  f | ⇑   ∨
    A ---------> C
    |     | hB   |
    | ⇗   ∨   ⇗  |
 hA |     B'     | hC
    | h' ∧  \ g' |
    |  /  ⇑   \  |
    ∨/          ∨∨
    A' --------> C'
          f'

we have commuting prisms of precomposition maps

                    (B' → S)
         (- ∘ g') ∧     |     \ (- ∘ h')
                /       |       \
              / (- ∘ f')| ⇑       ∨
       (C' → S) ---------------> (A' → S)
           |            |            |
           |            | (- ∘ hB)   |
           |     ⇙      ∨      ⇙     |
  (- ∘ hC) |         (B → S)         | (- ∘ hA)
           |  (- ∘ g) ∧   \ (- ∘ h)  |
           |       /    ⇑    \       |
           ∨    /               ∨    ∨
        (C → S) ----------------> (A → S).
                     (- ∘ f)

Observe that the bottom and top triangles have switched positions, the diagram is mirrored along the vertical axis compared to the underlying commuting prism, and that the direction of the homotopies of the vertical squares are flipped.

module _
  { l1 l2 l3 l1' l2' l3' l : Level}
  { A : UU l1} {B : UU l2} {C : UU l3}
  ( f : A → C) (g : B → C) (h : A → B)
  { A' : UU l1'} {B' : UU l2'} {C' : UU l3'}
  ( f' : A' → C') (g' : B' → C') (h' : A' → B')
  ( hA : A → A') (hB : B → B') (hC : C → C')
  ( top : coherence-triangle-maps f g h)
  ( front : coherence-square-maps f hA hC f')
  ( right : coherence-square-maps g hB hC g')
  ( left : coherence-square-maps h hA hB h')
  ( bottom : coherence-triangle-maps f' g' h')
  ( H :
    vertical-coherence-prism-maps f g h f' g' h' hA hB hC
      ( top)
      ( front)
      ( right)
      ( left)
      ( bottom))
  ( S : UU l)
  where

  precomp-vertical-coherence-prism-inv-squares-maps :
    vertical-coherence-prism-inv-squares-maps
      ( precomp f' S)
      ( precomp h' S)
      ( precomp g' S)
      ( precomp f S)
      ( precomp h S)
      ( precomp g S)
      ( precomp hC S)
      ( precomp hB S)
      ( precomp hA S)
      ( precomp-coherence-triangle-maps f' g' h' bottom S)
      ( precomp-coherence-square-maps f hA hC f' front S)
      ( precomp-coherence-square-maps h hA hB h' left S)
      ( precomp-coherence-square-maps g hB hC g' right S)
      ( precomp-coherence-triangle-maps f g h top S)
  precomp-vertical-coherence-prism-inv-squares-maps =
    ( ap-concat-htpy
      ( htpy-precomp front S)
      ( inv-distributive-htpy-precomp-left-whisker hC top S)) ∙h
    ( inv-htpy
      ( distributive-htpy-precomp-concat-htpy front (hC ·l top) S)) ∙h
    ( λ i → ap eq-htpy (ap (i ·l_) (eq-htpy (inv-htpy H)))) ∙h
    ( distributive-htpy-precomp-concat-htpy
      ( bottom ·r hA)
      ( pasting-horizontal-coherence-square-maps h g hA hB hC h' g' left right)
      ( S)) ∙h
    ( ap-binary-concat-htpy
      ( distributive-htpy-precomp-right-whisker hA bottom S)
      ( ( distributive-htpy-precomp-concat-htpy (g' ·l left) (right ·r h) S) ∙h
        ( ap-binary-concat-htpy
          ( distributive-htpy-precomp-left-whisker g' left S)
          ( distributive-htpy-precomp-right-whisker h right S))))

  precomp-vertical-coherence-prism-maps :
    vertical-coherence-prism-maps
      ( precomp f' S)
      ( precomp h' S)
      ( precomp g' S)
      ( precomp f S)
      ( precomp h S)
      ( precomp g S)
      ( precomp hC S)
      ( precomp hB S)
      ( precomp hA S)
      ( precomp-coherence-triangle-maps f' g' h' bottom S)
      ( inv-htpy (precomp-coherence-square-maps f hA hC f' front S))
      ( inv-htpy (precomp-coherence-square-maps h hA hB h' left S))
      ( inv-htpy (precomp-coherence-square-maps g hB hC g' right S))
      ( precomp-coherence-triangle-maps f g h top S)
  precomp-vertical-coherence-prism-maps =
    vertical-coherence-prism-maps-vertical-coherence-prism-inv-squares-maps
      ( precomp f' S)
      ( precomp h' S)
      ( precomp g' S)
      ( precomp f S)
      ( precomp h S)
      ( precomp g S)
      ( precomp hC S)
      ( precomp hB S)
      ( precomp hA S)
      ( precomp-coherence-triangle-maps f' g' h' bottom S)
      ( precomp-coherence-square-maps f hA hC f' front S)
      ( precomp-coherence-square-maps h hA hB h' left S)
      ( precomp-coherence-square-maps g hB hC g' right S)
      ( precomp-coherence-triangle-maps f g h top S)
      ( precomp-vertical-coherence-prism-inv-squares-maps)

module _
  { l1 l2 l3 l1' l2' l3' l : Level}
  { A : UU l1} {B : UU l2} {C : UU l3}
  ( f : A → C) (g : B → C) (h : A → B)
  { A' : UU l1'} {B' : UU l2'} {C' : UU l3'}
  ( f' : A' → C') (g' : B' → C') (h' : A' → B')
  ( hA : A → A') (hB : B → B') (hC : C → C')
  ( inv-top : coherence-triangle-maps' f g h)
  ( front : coherence-square-maps f hA hC f')
  ( right : coherence-square-maps g hB hC g')
  ( left : coherence-square-maps h hA hB h')
  ( inv-bottom : coherence-triangle-maps' f' g' h')
  ( H :
    vertical-coherence-prism-inv-triangles-maps f g h f' g' h' hA hB hC
      ( inv-top)
      ( front)
      ( right)
      ( left)
      ( inv-bottom))
  (S : UU l)
  where

  precomp-vertical-inv-boundary-vertical-coherence-inv-triangles-prism-maps :
    vertical-coherence-prism-inv-boundary-maps
      ( precomp f' S)
      ( precomp h' S)
      ( precomp g' S)
      ( precomp f S)
      ( precomp h S)
      ( precomp g S)
      ( precomp hC S)
      ( precomp hB S)
      ( precomp hA S)
      ( precomp-coherence-triangle-maps' f' g' h' inv-bottom S)
      ( precomp-coherence-square-maps f hA hC f' front S)
      ( precomp-coherence-square-maps h hA hB h' left S)
      ( precomp-coherence-square-maps g hB hC g' right S)
      ( precomp-coherence-triangle-maps' f g h inv-top S)
  precomp-vertical-inv-boundary-vertical-coherence-inv-triangles-prism-maps =
    ( inv-htpy
      ( ( compute-concat-htpy-precomp
          ( (g' ·l left) ∙h (right ·r h))
          ( hC ·l inv-top)
          ( S)) ∙h
        ( ap-binary-concat-htpy
          ( ( compute-concat-htpy-precomp (g' ·l left) (right ·r h) S) ∙h
            ( ap-binary-concat-htpy
              ( distributive-htpy-precomp-left-whisker g' left S)
              ( distributive-htpy-precomp-right-whisker h right S)))
          ( distributive-htpy-precomp-left-whisker hC inv-top S)))) ∙h
    ( λ i → ap (λ p → htpy-precomp p S i) (eq-htpy H)) ∙h
    ( compute-concat-htpy-precomp (inv-bottom ·r hA) front S) ∙h
    ( ap-concat-htpy'
      ( htpy-precomp front S)
      ( distributive-htpy-precomp-right-whisker hA inv-bottom S))

  precomp-vertical-coherence-prism-inv-triangles-maps :
    vertical-coherence-prism-inv-triangles-maps
      ( precomp f' S)
      ( precomp h' S)
      ( precomp g' S)
      ( precomp f S)
      ( precomp h S)
      ( precomp g S)
      ( precomp hC S)
      ( precomp hB S)
      ( precomp hA S)
      ( precomp-coherence-triangle-maps' f' g' h' inv-bottom S)
      ( inv-htpy (precomp-coherence-square-maps f hA hC f' front S))
      ( inv-htpy (precomp-coherence-square-maps h hA hB h' left S))
      ( inv-htpy (precomp-coherence-square-maps g hB hC g' right S))
      ( precomp-coherence-triangle-maps' f g h inv-top S)
  precomp-vertical-coherence-prism-inv-triangles-maps =
    vertical-coherence-prism-inv-triangles-maps-vertical-coherence-prism-inv-boundary-maps
      ( precomp f' S)
      ( precomp h' S)
      ( precomp g' S)
      ( precomp f S)
      ( precomp h S)
      ( precomp g S)
      ( precomp hC S)
      ( precomp hB S)
      ( precomp hA S)
      ( precomp-coherence-triangle-maps' f' g' h' inv-bottom S)
      ( precomp-coherence-square-maps f hA hC f' front S)
      ( precomp-coherence-square-maps h hA hB h' left S)
      ( precomp-coherence-square-maps g hB hC g' right S)
      ( precomp-coherence-triangle-maps' f g h inv-top S)
      ( precomp-vertical-inv-boundary-vertical-coherence-inv-triangles-prism-maps)

Commuting prisms of maps induce commuting prisms of postcomposition maps

Given a commuting prism of maps

          B
      h  ∧| \  g
       /  |   \
     /  f | ⇑   ∨
    A ---------> C
    |     | hB   |
    | ⇗   ∨   ⇗  |
 hA |     B'     | hC
    | h' ∧  \ g' |
    |  /  ⇑   \  |
    ∨/          ∨∨
    A' --------> C'
          f'

we have commuting prisms of postcomposition mapss

                     (S → B)
          (h ∘ -) ∧     |     \ (g ∘ -)
                /       |       \
              /  (f ∘ -)| ⇑       ∨
        (S → A) ----------------> (S → C)
           |            |            |
           |            | (hB ∘ -)   |
           |     ⇗      ∨      ⇗     |
  (hA ∘ -) |         (S → B')        | (hC ∘ -)
           | (h' ∘ -) ∧   \ (g' ∘ -) |
           |       /    ⇑    \       |
           ∨    /               ∨    ∨
        (S → A') ---------------> (S → C').
                     (f' ∘ -)

module _
  { l1 l2 l3 l1' l2' l3' l : Level}
  { A : UU l1} {B : UU l2} {C : UU l3}
  ( f : A → C) (g : B → C) (h : A → B)
  { A' : UU l1'} {B' : UU l2'} {C' : UU l3'}
  ( f' : A' → C') (g' : B' → C') (h' : A' → B')
  ( hA : A → A') (hB : B → B') (hC : C → C')
  ( inv-top : coherence-triangle-maps' f g h)
  ( front : coherence-square-maps f hA hC f')
  ( right : coherence-square-maps g hB hC g')
  ( left : coherence-square-maps h hA hB h')
  ( inv-bottom : coherence-triangle-maps' f' g' h')
  ( H :
    vertical-coherence-prism-inv-triangles-maps f g h f' g' h' hA hB hC
      ( inv-top)
      ( front)
      ( right)
      ( left)
      ( inv-bottom))
  (S : UU l)
  where

  postcomp-vertical-coherence-prism-inv-triangles-maps :
    vertical-coherence-prism-inv-triangles-maps
      ( postcomp S f)
      ( postcomp S g)
      ( postcomp S h)
      ( postcomp S f')
      ( postcomp S g')
      ( postcomp S h')
      ( postcomp S hA)
      ( postcomp S hB)
      ( postcomp S hC)
      ( htpy-postcomp S inv-top)
      ( htpy-postcomp S front)
      ( htpy-postcomp S right)
      ( htpy-postcomp S left)
      ( htpy-postcomp S inv-bottom)
  postcomp-vertical-coherence-prism-inv-triangles-maps =
    ( inv-htpy
      ( ( distributive-htpy-postcomp-concat-htpy
          ( g' ·l left ∙h right ·r h)
          ( hC ·l inv-top)
          ( S)) ∙h
        ( ap-binary-concat-htpy
          ( ( distributive-htpy-postcomp-concat-htpy
              ( g' ·l left)
              ( right ·r h) S) ∙h
            ( ap-binary-concat-htpy
              ( distributive-htpy-postcomp-left-whisker g' left S)
              ( distributive-htpy-postcomp-right-whisker h right S)))
          ( distributive-htpy-postcomp-left-whisker hC inv-top S)))) ∙h
    ( λ i → ap (λ p → htpy-postcomp S p i) (eq-htpy H)) ∙h
    ( distributive-htpy-postcomp-concat-htpy (inv-bottom ·r hA) front S)