Commuting squares of homotopies
Content created by Egbert Rijke, Fredrik Bakke and Raymond Baker.
Created on 2024-02-06.
Last modified on 2024-07-23.
module foundation-core.commuting-squares-of-homotopies where
Imports
open import foundation.universe-levels open import foundation.whiskering-homotopies-composition open import foundation-core.commuting-squares-of-identifications open import foundation-core.homotopies open import foundation-core.whiskering-homotopies-concatenation
Idea
A square of homotopies
top
f ------> g
| |
left | | right
∨ ∨
h ------> i
bottom
is said to be a commuting square¶ of
homotopies if there is a homotopy left ∙h bottom ~ top ∙h right . Such a
homotopy is called a
coherence¶
of the square.
Definitions
Commuting squares of homotopies
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) where coherence-square-homotopies : UU (l1 ⊔ l2) coherence-square-homotopies = left ∙h bottom ~ top ∙h right coherence-square-homotopies' : UU (l1 ⊔ l2) coherence-square-homotopies' = top ∙h right ~ left ∙h bottom
Horizontally constant squares
Horizontally constant squares¶ are commuting squares of homotopies of the form
refl-htpy
f ----------> f
| |
H | | H
∨ ∨
g ----------> g.
refl-htpy
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g : (x : A) → B x} (H : f ~ g) where horizontal-refl-coherence-square-homotopies : coherence-square-homotopies refl-htpy H H refl-htpy horizontal-refl-coherence-square-homotopies x = horizontal-refl-coherence-square-identifications (H x)
Vertically constant squares
Vertically constant squares¶ are commuting squares of homotopies of the form
H
f -----> g
| |
refl-htpy | | refl-htpy
∨ ∨
f -----> g.
H
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g : (x : A) → B x} (H : f ~ g) where vertical-refl-coherence-square-homotopies : coherence-square-homotopies H refl-htpy refl-htpy H vertical-refl-coherence-square-homotopies x = vertical-refl-coherence-square-identifications (H x)
Squares with refl on the top and bottom
Given a homotopy H ~ H', we can obtain a coherence square with refl-htpy on
the top and bottom.
refl-htpy
f ----------> f
| |
H | | H'
∨ ∨
g ----------> g
refl-htpy
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g : (x : A) → B x} (H H' : f ~ g) where coherence-square-homotopies-horizontal-refl : H ~ H' → coherence-square-homotopies ( refl-htpy) ( H) ( H') ( refl-htpy) coherence-square-homotopies-horizontal-refl K = right-unit-htpy ∙h K
Conversely, given a coherence square as above, we can obtain a homotopy
H ~ H'.
inv-coherence-square-homotopies-horizontal-refl : coherence-square-homotopies ( refl-htpy) ( H) ( H') ( refl-htpy) → H ~ H' inv-coherence-square-homotopies-horizontal-refl K = inv-htpy-right-unit-htpy ∙h K
Squares with refl-htpy on the left and right
Given a homotopy H ~ H', we can obtain a coherence square with refl-htpy on
the left and right.
H'
f ------> g
| |
refl-htpy | | refl-htpy
∨ ∨
f ------> g
H
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g : (x : A) → B x} (H H' : f ~ g) where coherence-square-homotopies-vertical-refl : H ~ H' → coherence-square-homotopies ( H') ( refl-htpy) ( refl-htpy) ( H) coherence-square-homotopies-vertical-refl K = K ∙h inv-htpy right-unit-htpy
Conversely, given a coherence square as above, we can obtain a homotopy
H ~ H'.
inv-coherence-square-homotopies-vertical-refl : coherence-square-homotopies ( H') ( refl-htpy) ( refl-htpy) ( H) → H ~ H' inv-coherence-square-homotopies-vertical-refl K = K ∙h right-unit-htpy
Operations
Inverting squares of homotopies horizontally
Given a commuting square of homotopies
top
f -------> g
| |
left | | right
∨ ∨
h -------> i,
bottom
the square of homotopies
top⁻¹
g ------------> f
| |
right | | left
∨ ∨
i ------------> h
bottom⁻¹
commutes.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} where horizontal-inv-coherence-square-homotopies : (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → coherence-square-homotopies top left right bottom → coherence-square-homotopies (inv-htpy top) right left (inv-htpy bottom) horizontal-inv-coherence-square-homotopies top left right bottom H x = horizontal-inv-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x)
Inverting squares of homotopies vertically
Given a commuting square of homotopies
top
f -------> g
| |
left | | right
∨ ∨
h -------> i,
bottom
the square of homotopies
bottom
h -------> i
| |
left⁻¹ | | right⁻¹
∨ ∨
f -------> g
top
commutes.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} where vertical-inv-coherence-square-homotopies : (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → coherence-square-homotopies top left right bottom → coherence-square-homotopies bottom (inv-htpy left) (inv-htpy right) top vertical-inv-coherence-square-homotopies top left right bottom H x = vertical-inv-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x)
Functions acting on squares of homotopies
Given a commuting square of homotopies
top
f -------> g
| |
left | | right
∨ ∨
h -------> i
bottom
in (x : A) → B x, and given a dependent map F : {x : A} → B x → C x, the
square of homotopies
F ·l top
f f -----------> f g
| |
F ·l left | | F ·l right
∨ ∨
h -------------> i
F ·l bottom
commutes.
module _ {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} {f g h i : (x : A) → B x} (F : {x : A} → B x → C x) where map-coherence-square-homotopies : (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → coherence-square-homotopies top left right bottom → coherence-square-homotopies ( F ·l top) ( F ·l left) ( F ·l right) ( F ·l bottom) map-coherence-square-homotopies top left right bottom H x = map-coherence-square-identifications ( F) ( top x) ( left x) ( right x) ( bottom x) ( H x)
Similarly we may whisker it on the right.
module _ {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : B → UU l3} {f g h i : (y : B) → C y} where right-whisker-comp-coherence-square-homotopies : (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → (F : A → B) → coherence-square-homotopies top left right bottom → coherence-square-homotopies ( top ·r F) ( left ·r F) ( right ·r F) ( bottom ·r F) right-whisker-comp-coherence-square-homotopies top left right bottom F α = α ·r F
Concatenating homotopies of edges and coherences of commuting squares of homotopies
Consider a commuting square of homotopies and a homotopy of one of the four sides with another homotopy, as for example in the diagram below:
top
a ---------> b
| | |
left | right |~| right'
∨ ∨ ∨
c ---------> d.
bottom
Then any homotopy witnessing that the square commutes can be concatenated with the homotopy on the side, to obtain a new commuting square of homotopies.
Note. To avoid cyclic module dependencies we will give direct proofs that concatenating homotopies of edges of a square with the coherence of its commutativity is an equivalence.
Concatenating homotopies of the top edge with a coherence of a commuting square of homotopies
Consider a commuting diagram of homotopies
top'
------->
f -------> g
| top |
left | | right
∨ ∨
h -------> i.
bottom
with a homotopy top ~ top'. Then we get maps back and forth
top top'
f -------> g f -------> g
| | | |
left | | right ↔ left | | right
∨ ∨ ∨ ∨
h -------> i h -------> i.
bottom bottom
We record that this construction is an equivalence in
foundation.commuting-squares-of-homotopies.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) {top' : f ~ g} (s : top ~ top') where concat-top-homotopy-coherence-square-homotopies : coherence-square-homotopies top left right bottom → coherence-square-homotopies top' left right bottom concat-top-homotopy-coherence-square-homotopies H x = concat-top-identification-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( s x) ( H x) inv-concat-top-homotopy-coherence-square-homotopies : coherence-square-homotopies top' left right bottom → coherence-square-homotopies top left right bottom inv-concat-top-homotopy-coherence-square-homotopies H x = inv-concat-top-identification-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( s x) ( H x)
Concatenating homotopies of the left edge with a coherence of a commuting square of homotopies
Consider a commuting diagram of homotopies
top
f -------> g
| | |
left' | | left | right
∨ ∨ ∨
h -------> i.
bottom
with a homotopy left ~ left'. Then we get maps back and forth
top top
f -------> g f -------> g
| | | |
left | | right ↔ left' | | right
∨ ∨ ∨ ∨
h -------> i h -------> i.
bottom bottom
We record that this construction is an equivalence in
foundation.commuting-squares-of-homotopies.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) {left' : f ~ h} (s : left ~ left') where concat-left-homotopy-coherence-square-homotopies : coherence-square-homotopies top left right bottom → coherence-square-homotopies top left' right bottom concat-left-homotopy-coherence-square-homotopies H x = concat-left-identification-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( s x) ( H x) inv-concat-left-homotopy-coherence-square-homotopies : coherence-square-homotopies top left' right bottom → coherence-square-homotopies top left right bottom inv-concat-left-homotopy-coherence-square-homotopies H x = inv-concat-left-identification-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( s x) ( H x)
Concatenating homotopies of the right edge with a coherence of a commuting square of homotopies
Consider a commuting diagram of homotopies
top
f -------> g
| | |
left | right | | right'
∨ ∨ ∨
h -------> i.
bottom
with a homotopy right ~ right'. Then we get maps back and forth
top top
f -------> g f -------> g
| | | |
left | | right ↔ left | | right'
∨ ∨ ∨ ∨
h -------> i h -------> i.
bottom bottom
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) {right' : g ~ i} (s : right ~ right') where concat-right-homotopy-coherence-square-homotopies : coherence-square-homotopies top left right bottom → coherence-square-homotopies top left right' bottom concat-right-homotopy-coherence-square-homotopies H x = concat-right-identification-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( s x) ( H x) inv-concat-right-homotopy-coherence-square-homotopies : coherence-square-homotopies top left right' bottom → coherence-square-homotopies top left right bottom inv-concat-right-homotopy-coherence-square-homotopies H x = inv-concat-right-identification-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( s x) ( H x)
We record that this construction is an equivalence in
foundation.commuting-squares-of-homotopies.
Concatenating homotopies of the bottom edge with a coherence of a commuting square of homotopies
Consider a commuting diagram of homotopies
top
f -------> g
| |
left | | right
∨ bottom ∨
h -------> i.
------->
bottom'
with a homotopy bottom ~ bottom'. Then we get maps back and forth
top top
f -------> g f -------> g
| | | |
left | | right ↔ left | | right
∨ ∨ ∨ ∨
h -------> i h -------> i.
bottom bottom'
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) {bottom' : h ~ i} (s : bottom ~ bottom') where concat-bottom-homotopy-coherence-square-homotopies : coherence-square-homotopies top left right bottom → coherence-square-homotopies top left right bottom' concat-bottom-homotopy-coherence-square-homotopies H x = concat-bottom-identification-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( s x) ( H x) inv-concat-bottom-homotopy-coherence-square-homotopies : coherence-square-homotopies top left right bottom' → coherence-square-homotopies top left right bottom inv-concat-bottom-homotopy-coherence-square-homotopies H x = inv-concat-bottom-identification-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( s x) ( H x)
We record that this construction is an equivalence in
foundation.commuting-squares-of-homotopies.
Whiskering and splicing coherences of commuting squares of homotopies with respect to concatenation of homotopies
Given a commuting square of homotopies
top
f -------> g
| |
left | | right
∨ ∨
h -------> i,
bottom
we may consider four ways of attaching new homotopies to it:
-
Prepending
H : u ~ fto the left gives us a commuting squareH ∙h top u -------> g | | H ∙h left | | right ∨ ∨ h -------> i. bottomMore precisely, we have an equivalence
(left ∙h bottom ~ top ∙h right) ≃ ((H ∙h left) ∙h bottom ~ (H ∙h top) ∙h right). -
Appending a homotopy
H : i ~ uto the right gives a commuting square of homotopiestop f ------------> g | | left | | right ∙h H ∨ ∨ h ------------> u. bottom ∙h HMore precisely, we have an equivalence
(left ∙h bottom ~ top ∙h right) ≃ (left ∙h (bottom ∙h H) ~ top ∙h (right ∙h H)). -
Splicing a homotopy
H : h ~ uand its inverse into the middle gives a commuting square of homotopiestop f --------------> g | | left ∙h H | | right ∨ ∨ u --------------> i. H⁻¹ ∙h bottomMore precisely, we have an equivalence
(left ∙h bottom ~ top ∙h right) ≃ ((left ∙h H) ∙h (H⁻¹ ∙h bottom) ~ top ∙h right).Similarly, we have an equivalence
(left ∙h bottom ~ top ∙h right) ≃ ((left ∙h H⁻¹) ∙h (H ∙h bottom) ~ top ∙h right). -
Splicing a homotopy
H : g ~ uand its inverse into the middle gives a commuting square of homotopiestop ∙h H f --------> u | | left | | H⁻¹ ∙h right ∨ ∨ h --------> i. bottomMore precisely, we have an equivalence
(left ∙h bottom ~ top ∙h right) ≃ (left ∙h bottom ~ (top ∙h H) ∙h (H⁻¹ ∙h right)).Similarly, we have an equivalence
(left ∙h bottom ~ top ∙h right) ≃ (left ∙h bottom ~ (top ∙h H⁻¹) ∙h (H ∙h right)).
These operations are useful in proofs involving homotopy algebra, because taking
equiv-right-whisker-concat-coherence-square-homotopies as an example, it
provides us with two maps: the forward direction states
(H ∙h r ~ K ∙h s) → (H ∙h (r ∙h t)) ~ K ∙h (s ∙h t)), which allows one to
append a homotopy without needing to reassociate on the right, and the backwards
direction conversely allows one to cancel out a homotopy in parentheses.
Left whiskering coherences of commuting squares of homotopies
For any homotopy H : u ~ f we obtain maps back and forth
top H ∙h top
f -------> g u -------> g
| | | |
left | | right ↔ H ∙h left | | right
∨ ∨ ∨ ∨
h -------> i h -------> i
bottom bottom
of coherences of commuting squares of homotopies. We show in
foundation.commuting-squares-of-homotopies
that these maps are equivalences.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} where left-whisker-concat-coherence-square-homotopies : {u : (x : A) → B x} (H : u ~ f) (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → coherence-square-homotopies top left right bottom → coherence-square-homotopies (H ∙h top) (H ∙h left) right bottom left-whisker-concat-coherence-square-homotopies H top left right bottom coh x = left-whisker-concat-coherence-square-identifications ( H x) ( top x) ( left x) ( right x) ( bottom x) ( coh x) left-unwhisker-concat-coherence-square-homotopies : {u : (x : A) → B x} (H : u ~ f) (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → coherence-square-homotopies (H ∙h top) (H ∙h left) right bottom → coherence-square-homotopies top left right bottom left-unwhisker-concat-coherence-square-homotopies H top left right bottom coh x = left-unwhisker-concat-coherence-square-identifications ( H x) ( top x) ( left x) ( right x) ( bottom x) ( coh x)
Right whiskering coherences of commuting squares of homotopies
For any homotopy H : i ~ u we obtain maps back and forth
top top
f -------> g f ------------> g
| | | |
left | | right ↔ left | | right ∙h H
∨ ∨ ∨ ∨
h -------> i h ------------> i
bottom bottom ∙h H
of coherences of commuting squares of homotopies. We show in
foundation.commuting-squares-of-homotopies
that these maps are equivalences.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) where right-whisker-concat-coherence-square-homotopies : coherence-square-homotopies top left right bottom → {u : (x : A) → B x} (H : i ~ u) → coherence-square-homotopies top left (right ∙h H) (bottom ∙h H) right-whisker-concat-coherence-square-homotopies coh H x = right-whisker-concat-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( coh x) ( H x) right-unwhisker-cohernece-square-homotopies : {u : (x : A) → B x} (H : i ~ u) → coherence-square-homotopies top left (right ∙h H) (bottom ∙h H) → coherence-square-homotopies top left right bottom right-unwhisker-cohernece-square-homotopies H coh x = right-unwhisker-cohernece-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x) ( coh x)
Double whiskering of commuting squares of homotopies
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h u v i : (x : A) → B x} where double-whisker-coherence-square-homotopies : (p : f ~ g) (top : g ~ u) (left : g ~ h) (right : u ~ v) (bottom : h ~ v) (s : v ~ i) → coherence-square-homotopies top left right bottom → coherence-square-homotopies ( p ∙h top) ( p ∙h left) ( right ∙h s) ( bottom ∙h s) double-whisker-coherence-square-homotopies p top left right bottom q H = left-whisker-concat-coherence-square-homotopies p top left ( right ∙h q) ( bottom ∙h q) ( right-whisker-concat-coherence-square-homotopies ( top) ( left) ( right) ( bottom) ( H) ( q))
Left splicing coherences of commuting squares of homotopies
For any inverse pair of homotopies H : g ~ u and K : u ~ g equipped with
α : inv-htpy H ~ K we obtain maps back and forth
top top
f -------> g f -----------> g
| | | |
left | | right ↔ left ∙h H | | right
∨ ∨ ∨ ∨
h -------> i u -----------> i
bottom K ∙h bottom
of coherences of commuting squares of homotopies. We show in
foundation.commuting-squares-of-homotopies
that these maps are equivalences.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) where left-splice-coherence-square-homotopies : {u : (x : A) → B x} (H : h ~ u) (K : u ~ h) (α : inv-htpy H ~ K) → coherence-square-homotopies top left right bottom → coherence-square-homotopies top (left ∙h H) right (K ∙h bottom) left-splice-coherence-square-homotopies H K α coh x = left-splice-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x) ( K x) ( α x) ( coh x) left-unsplice-coherence-square-homotopies : {u : (x : A) → B x} (H : h ~ u) (K : u ~ h) (α : inv-htpy H ~ K) → coherence-square-homotopies top (left ∙h H) right (K ∙h bottom) → coherence-square-homotopies top left right bottom left-unsplice-coherence-square-homotopies H K α coh x = left-unsplice-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x) ( K x) ( α x) ( coh x)
Right splicing coherences of commuting squares of homotopies
For any inverse pair of homotopies H : g ~ u and K : u ~ g equipped with
α : inv-htpy H ~ K we obtain maps back and forth
top top ∙h H
f -------> g f --------> u
| | | |
left | | right ↔ left | | K ∙h right
∨ ∨ ∨ ∨
h -------> i h --------> i
bottom bottom
of coherences of commuting squares of homotopies. We show in
foundation.commuting-squares-of-homotopies
that these maps are equivalences.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) where right-splice-coherence-square-homotopies : {u : (x : A) → B x} (H : g ~ u) (K : u ~ g) (α : inv-htpy H ~ K) → coherence-square-homotopies top left right bottom → coherence-square-homotopies (top ∙h H) left (inv-htpy H ∙h right) bottom right-splice-coherence-square-homotopies H K α coh x = right-splice-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x) ( K x) ( α x) ( coh x) right-unsplice-coherence-square-homotopies : {u : (x : A) → B x} (H : g ~ u) (K : u ~ g) (α : inv-htpy H ~ K) → coherence-square-homotopies (top ∙h H) left (inv-htpy H ∙h right) bottom → coherence-square-homotopies top left right bottom right-unsplice-coherence-square-homotopies H K α coh x = right-unsplice-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x) ( K x) ( α x) ( coh x)
Horizontally pasting squares of homotopies
Consider two squares of homotopies as in the diagram
top-left top-right
a -------------> b -------------> c
| | |
left | | middle | right
∨ ∨ ∨
d -------------> e -------------> f
bottom-left bottom-right
with H : left ∙h bottom-left ~ top-left ∙h middle and
K : middle ∙h bottom-right ~ top-right ∙h right. Then the outer square
commutes.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a b c d e f : (x : A) → B x} (top-left : a ~ b) (top-right : b ~ c) (left : a ~ d) (middle : b ~ e) (right : c ~ f) (bottom-left : d ~ e) (bottom-right : e ~ f) where horizontal-pasting-coherence-square-homotopies : coherence-square-homotopies top-left left middle bottom-left → coherence-square-homotopies top-right middle right bottom-right → coherence-square-homotopies (top-left ∙h top-right) left right (bottom-left ∙h bottom-right) horizontal-pasting-coherence-square-homotopies H K x = horizontal-pasting-coherence-square-identifications ( top-left x) ( top-right x) ( left x) ( middle x) ( right x) ( bottom-left x) ( bottom-right x) ( H x) ( K x)
Vertically pasting squares of homotopies
Consider two squares of homotopies as in the diagram
top
a --------> b
| |
top-left | | top-right
∨ middle ∨
c --------> d
| |
bottom-left | | bottom-right
∨ ∨
e --------> f
bottom
with H : top-left ∙h middle ~ top ∙h top-right and
K : bottom-left ∙h bottom ~ middle ∙h bottom-right. Then the outer square
commutes.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a b c d e f : (x : A) → B x} (top : a ~ b) (top-left : a ~ c) (top-right : b ~ d) (middle : c ~ d) (bottom-left : c ~ e) (bottom-right : d ~ f) (bottom : e ~ f) where vertical-pasting-coherence-square-homotopies : coherence-square-homotopies top top-left top-right middle → coherence-square-homotopies middle bottom-left bottom-right bottom → coherence-square-homotopies top (top-left ∙h bottom-left) (top-right ∙h bottom-right) bottom vertical-pasting-coherence-square-homotopies H K x = vertical-pasting-coherence-square-identifications ( top x) ( top-left x) ( top-right x) ( middle x) ( bottom-left x) ( bottom-right x) ( bottom x) ( H x) ( K x)
Properties
Left unit law for horizontal pasting of commuting squares of homotopies
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} where left-unit-law-horizontal-pasting-coherence-square-homotopies : (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → (H : coherence-square-homotopies top left right bottom) → horizontal-pasting-coherence-square-homotopies ( refl-htpy) ( top) ( left) ( left) ( right) ( refl-htpy) ( bottom) ( horizontal-refl-coherence-square-homotopies left) ( H) ~ H left-unit-law-horizontal-pasting-coherence-square-homotopies top left right bottom H x = left-unit-law-horizontal-pasting-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x)
Right unit law for horizontal pasting of commuting squares of homotopies
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} where right-unit-law-horizontal-pasting-coherence-square-homotopies : (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → (H : coherence-square-homotopies top left right bottom) → horizontal-pasting-coherence-square-homotopies ( top) ( refl-htpy) ( left) ( right) ( right) ( bottom) ( refl-htpy) ( H) ( horizontal-refl-coherence-square-homotopies right) ∙h right-whisker-concat-htpy right-unit-htpy right ~ left-whisker-concat-htpy left right-unit-htpy ∙h H right-unit-law-horizontal-pasting-coherence-square-homotopies top left right bottom H x = right-unit-law-horizontal-pasting-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x)
Left unit law for vertical pasting of commuting squares of homotopies
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} where left-unit-law-vertical-pasting-coherence-square-homotopies : (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → (H : coherence-square-homotopies top left right bottom) → vertical-pasting-coherence-square-homotopies ( top) ( refl-htpy) ( refl-htpy) ( top) ( left) ( right) ( bottom) ( vertical-refl-coherence-square-homotopies top) ( H) ~ H left-unit-law-vertical-pasting-coherence-square-homotopies top left right bottom H x = left-unit-law-vertical-pasting-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x)
Right unit law for vertical pasting of commuting squares of homotopies
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h i : (x : A) → B x} where right-unit-law-vertical-pasting-coherence-square-homotopies : (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) → (H : coherence-square-homotopies top left right bottom) → vertical-pasting-coherence-square-homotopies ( top) ( left) ( right) ( bottom) ( refl-htpy) ( refl-htpy) ( bottom) ( H) ( vertical-refl-coherence-square-homotopies bottom) ∙h left-whisker-concat-htpy top right-unit-htpy ~ right-whisker-concat-htpy right-unit-htpy bottom ∙h H right-unit-law-vertical-pasting-coherence-square-homotopies top left right bottom H x = right-unit-law-vertical-pasting-coherence-square-identifications ( top x) ( left x) ( right x) ( bottom x) ( H x)
Computing the right whiskering of a vertically constant square with a homotopy
Consider the vertically constant square of homotopies
H
f -----> g
| |
refl | | refl
∨ ∨
f -----> g
H
at a homotopy H : f ~ g, and consider a homotopy K : g ~ h. Then the right
whiskering of the above square with K is the commuting square of homotopies
H
f -------> g
| |
refl | refl | K
∨ ∨
f -------> h
H ∙h K
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h : (x : A) → B x} where right-whisker-concat-vertical-refl-coherence-square-homotopies : (H : f ~ g) (K : g ~ h) → right-whisker-concat-coherence-square-homotopies H refl-htpy refl-htpy H ( vertical-refl-coherence-square-homotopies H) ( K) ~ refl-htpy right-whisker-concat-vertical-refl-coherence-square-homotopies H K x = right-whisker-concat-vertical-refl-coherence-square-identifications ( H x) ( K x)
Computing the right whiskering of a horizontally constant square with a homotopy
Consider a horizontally constant commuting square of homotopies
refl-htpy
f ----------> f
| |
H | | H
∨ ∨
g ----------> g
refl-htpy
at a homotopy H and consider a homotopy K : g ~ h. Then the right whiskering
of the above square with K is the square
refl-htpy
f ----------> f
| |
H | refl-htpy | H ∙h K
∨ ∨
g ----------> h.
K
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h : (x : A) → B x} where right-whisker-concat-horizontal-refl-coherence-square-homotopies : (H : f ~ g) (K : g ~ h) → right-whisker-concat-coherence-square-homotopies refl-htpy H H refl-htpy ( horizontal-refl-coherence-square-homotopies H) ( K) ~ refl-htpy right-whisker-concat-horizontal-refl-coherence-square-homotopies H K x = right-whisker-concat-horizontal-refl-coherence-square-identifications ( H x) ( K x)
Computing the left whiskering of a horizontally constant square with a homotopy
Consider a homotopy H : f ~ g and a horizontally constant commuting square of
homotopies
refl-htpy
g ----------> g
| |
K | | K
∨ ∨
h ----------> h
refl-htpy
at a homotopy K : g ~ h. The the left whiskering of the above square with H
is the commuting square
K ∙h refl-htpy
f -----------------------------------------------------------------> g
| |
K ∙h H | right-unit-htpy ∙h (right-whisker-concat-htpy right-unit-htpy H)⁻¹ | H
∨ ∨
h -----------------------------------------------------------------> h.
refl-htpy
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h : (x : A) → B x} where left-whisker-concat-horizontal-refl-coherence-square-homotopies : (H : f ~ g) (K : g ~ h) → left-whisker-concat-coherence-square-homotopies H refl-htpy K K refl-htpy ( horizontal-refl-coherence-square-homotopies K) ∙h right-whisker-concat-htpy right-unit-htpy K ~ right-unit-htpy left-whisker-concat-horizontal-refl-coherence-square-homotopies H K x = left-whisker-concat-horizontal-refl-coherence-square-identifications ( H x) ( K x) left-whisker-concat-horizontal-refl-coherence-square-homotopies' : (H : f ~ g) (K : g ~ h) → left-whisker-concat-coherence-square-homotopies H refl-htpy K K refl-htpy ( horizontal-refl-coherence-square-homotopies K) ~ right-unit-htpy ∙h inv-htpy (right-whisker-concat-htpy right-unit-htpy K) left-whisker-concat-horizontal-refl-coherence-square-homotopies' H K x = left-whisker-concat-horizontal-refl-coherence-square-identifications' ( H x) ( K x)
Computing the left whiskering of a vertically constant square with a homotopy
Consider the vertically constant square of homotopies
K
g -----> h
| |
refl-htpy | | refl-htpy
∨ ∨
g -----> h
K
at a homotopy K : g ~ h and consider a homotopy H : f ~ g. Then the left
whiskering of the above square with H is the square
H ∙h K
f ----------------------------------------------------------> h
| |
H ∙h refl-htpy | right-whisker-concat-htpy right-unit-htpy K ∙h right-unit⁻¹ | refl-htpy
∨ ∨
g ----------------------------------------------------------> h.
K
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g h : (x : A) → B x} where left-whisker-concat-vertical-refl-coherence-square-homotopies : (H : f ~ g) (K : g ~ h) → left-whisker-concat-coherence-square-homotopies H K refl-htpy refl-htpy K ( vertical-refl-coherence-square-homotopies K) ∙h right-unit-htpy ~ right-whisker-concat-htpy right-unit-htpy K left-whisker-concat-vertical-refl-coherence-square-homotopies H K x = left-whisker-concat-vertical-refl-coherence-square-identifications ( H x) ( K x) left-whisker-concat-vertical-refl-coherence-square-homotopies' : (H : f ~ g) (K : g ~ h) → left-whisker-concat-coherence-square-homotopies H K refl-htpy refl-htpy K ( vertical-refl-coherence-square-homotopies K) ~ right-whisker-concat-htpy right-unit-htpy K ∙h inv-htpy right-unit-htpy left-whisker-concat-vertical-refl-coherence-square-homotopies' H K x = left-whisker-concat-vertical-refl-coherence-square-identifications' ( H x) ( K x)
Left whiskering horizontal concatenations of squares with homotopies
Consider a commuting diagram of homotopies of the form
top-left top-right
a -------------> c -------------> e
| | |
left | | middle | right
∨ ∨ ∨
b -------------> d -------------> f
bottom-left bottom-right
and consider a homotopy H : f ~ a. Then the left whiskering of H and the
horizontal concatenation of coherences of commuting squares is up to
associativity the horizontal concatenation of the squares
H ∙h top-left top-right
u -------------> c -------------> e
| | |
H ∙h left | | middle | right
∨ ∨ ∨
b -------------> d -------------> f
bottom-left bottom-right
where the left square is the left whiskering of H and the original left
square.
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} where left-whisker-concat-horizontal-pasting-coherence-square-homotopies : {u a b c d e f : (x : A) → B x} (H : u ~ a) (top-left : a ~ c) (top-right : c ~ e) (left : a ~ b) (middle : c ~ d) (right : e ~ f) (bottom-left : b ~ d) (bottom-right : d ~ f) (l : coherence-square-homotopies top-left left middle bottom-left) (r : coherence-square-homotopies top-right middle right bottom-right) → left-whisker-concat-coherence-square-homotopies H ( top-left ∙h top-right) ( left) ( right) ( bottom-left ∙h bottom-right) ( horizontal-pasting-coherence-square-homotopies ( top-left) ( top-right) ( left) ( middle) ( right) ( bottom-left) ( bottom-right) ( l) ( r)) ~ horizontal-pasting-coherence-square-homotopies ( H ∙h top-left) ( top-right) ( H ∙h left) ( middle) ( right) ( bottom-left) ( bottom-right) ( left-whisker-concat-coherence-square-homotopies H ( top-left) ( left) ( middle) ( bottom-left) ( l)) ( r) ∙h right-whisker-concat-htpy ( assoc-htpy H top-left top-right) ( right) left-whisker-concat-horizontal-pasting-coherence-square-homotopies H top-left top-right left middle right bottom-left bottom-right l r x = left-whisker-concat-horizontal-pasting-coherence-square-identifications ( H x) ( top-left x) ( top-right x) ( left x) ( middle x) ( right x) ( bottom-left x) ( bottom-right x) ( l x) ( r x)
Left whiskering vertical concatenations of squares with homotopies
Consider two squares of homotopies as in the diagram
top
a --------> b
| |
top-left | | top-right
∨ middle ∨
c --------> d
| |
bottom-left | | bottom-right
∨ ∨
e --------> f
bottom
and consider a homotopy H : f ~ a. Then the left whiskering of H with the
vertical pasting of the two squares above is up to associativity the vertical
pasting of the squares
H ∙h top
u --------> b
| |
H ∙h top-left | | top-right
∨ middle ∨
c --------> d
| |
bottom-left | | bottom-right
∨ ∨
e --------> f.
bottom
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} where left-whisker-concat-vertical-concat-coherence-square-homotopies : {u a b c d e f : (x : A) → B x} (H : u ~ a) → (top : a ~ b) (top-left : a ~ c) (top-right : b ~ d) (middle : c ~ d) (bottom-left : c ~ e) (bottom-right : d ~ f) (bottom : e ~ f) (t : coherence-square-homotopies top top-left top-right middle) → (b : coherence-square-homotopies middle bottom-left bottom-right bottom) → right-whisker-concat-htpy (assoc-htpy H top-left bottom-left) bottom ∙h left-whisker-concat-coherence-square-homotopies H ( top) ( top-left ∙h bottom-left) ( top-right ∙h bottom-right) ( bottom) ( vertical-pasting-coherence-square-homotopies ( top) ( top-left) ( top-right) ( middle) ( bottom-left) ( bottom-right) ( bottom) ( t) ( b)) ~ vertical-pasting-coherence-square-homotopies ( H ∙h top) ( H ∙h top-left) ( top-right) ( middle) ( bottom-left) ( bottom-right) ( bottom) ( left-whisker-concat-coherence-square-homotopies H ( top) ( top-left) ( top-right) ( middle) ( t)) ( b) left-whisker-concat-vertical-concat-coherence-square-homotopies H top top-left top-right middle bottom-left bottom-right bottom t b x = left-whisker-concat-vertical-concat-coherence-square-identifications ( H x) ( top x) ( top-left x) ( top-right x) ( middle x) ( bottom-left x) ( bottom-right x) ( bottom x) ( t x) ( b x)
Right whiskering horizontal pastings of commuting squares of homotopies
Consider a commuting diagram of homotopies of the form
top-left top-right
a -------------> c -------------> e
| | |
left | | middle | right
∨ ∨ ∨
b -------------> d -------------> f
bottom-left bottom-right
and consider a homotopy K : f ~ g. Then the right whiskering of the horizontal
pasting of the squares above is up to associativity the horizontal pasting of
the squares
top-left top-right
a -------------> c ------------------> e
| | |
left | | middle | right ∙h K
∨ ∨ ∨
b -------------> d ------------------> g
bottom-left bottom-right ∙h K
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} where right-whisker-concat-horizontal-pasting-coherence-square-homotopies : {a b c d e f g : (x : A) → B x} (top-left : a ~ c) (top-right : c ~ e) (left : a ~ b) (middle : c ~ d) (right : e ~ f) (bottom-left : b ~ d) (bottom-right : d ~ f) (l : coherence-square-homotopies top-left left middle bottom-left) → (r : coherence-square-homotopies top-right middle right bottom-right) → (K : f ~ g) → right-whisker-concat-coherence-square-homotopies ( top-left ∙h top-right) ( left) ( right) ( bottom-left ∙h bottom-right) ( horizontal-pasting-coherence-square-homotopies ( top-left) ( top-right) ( left) ( middle) ( right) ( bottom-left) ( bottom-right) ( l) ( r)) ( K) ~ left-whisker-concat-htpy left (assoc-htpy bottom-left bottom-right K) ∙h horizontal-pasting-coherence-square-homotopies ( top-left) ( top-right) ( left) ( middle) ( right ∙h K) ( bottom-left) ( bottom-right ∙h K) ( l) ( right-whisker-concat-coherence-square-homotopies ( top-right) ( middle) ( right) ( bottom-right) ( r) ( K)) right-whisker-concat-horizontal-pasting-coherence-square-homotopies top-left top-right left middle right bottom-left bottom-right l r K x = right-whisker-concat-horizontal-pasting-coherence-square-identifications ( top-left x) ( top-right x) ( left x) ( middle x) ( right x) ( bottom-left x) ( bottom-right x) ( l x) ( r x) ( K x)
Right whiskering vertical concatenations of squares with homotopies
Consider two squares of homotopies as in the diagram
top
a --------> b
| |
top-left | | top-right
∨ middle ∨
c --------> d
| |
bottom-left | | bottom-right
∨ ∨
e --------> f
bottom
and consider a homotopy K : f ~ g. Then the right whiskering of the vertical
pasting of the two squares above with K is up to associativity the vertical
pasting of the squares
top
a ------------> b
| |
top-left | | top-right
∨ middle ∨
c ------------> d
| |
bottom-left | | bottom-right ∙h K
∨ ∨
e ------------> g.
bottom ∙h K
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} where right-whisker-concat-vertical-pasting-coherence-square-homotopies : {a b c d e f g : (x : A) → B x} (top : a ~ b) (top-left : a ~ c) (top-right : b ~ d) (middle : c ~ d) (bottom-left : c ~ e) (bottom-right : d ~ f) (bottom : e ~ f) (t : coherence-square-homotopies top top-left top-right middle) → (b : coherence-square-homotopies middle bottom-left bottom-right bottom) → (K : f ~ g) → right-whisker-concat-coherence-square-homotopies ( top) ( top-left ∙h bottom-left) ( top-right ∙h bottom-right) ( bottom) ( vertical-pasting-coherence-square-homotopies ( top) ( top-left) ( top-right) ( middle) ( bottom-left) ( bottom-right) ( bottom) ( t) ( b)) ( K) ∙h left-whisker-concat-htpy top (assoc-htpy top-right bottom-right K) ~ vertical-pasting-coherence-square-homotopies ( top) ( top-left) ( top-right) ( middle) ( bottom-left) ( bottom-right ∙h K) ( bottom ∙h K) ( t) ( right-whisker-concat-coherence-square-homotopies ( middle) ( bottom-left) ( bottom-right) ( bottom) ( b) ( K)) right-whisker-concat-vertical-pasting-coherence-square-homotopies top top-left top-right middle bottom-left bottom-right bottom t b K x = right-whisker-concat-vertical-pasting-coherence-square-identifications ( top x) ( top-left x) ( top-right x) ( middle x) ( bottom-left x) ( bottom-right x) ( bottom x) ( t x) ( b x) ( K x)
Recent changes
- 2024-07-23. Raymond Baker. Eckmann-Hilton Updates (#1133).
- 2024-04-25. Fredrik Bakke. chore: Fix arrowheads in character diagrams (#1124).
- 2024-03-13. Egbert Rijke. Refactoring pointed types (#1056).
- 2024-02-19. Fredrik Bakke. Additions for coherently invertible maps (#1024).
- 2024-02-06. Egbert Rijke and Fredrik Bakke. Refactor files about identity types and homotopies (#1014).