Functoriality of sequential colimits

Content created by Fredrik Bakke, Vojtěch Štěpančík and Egbert Rijke.

Created on 2023-12-05.
Last modified on 2024-04-25.

module synthetic-homotopy-theory.functoriality-sequential-colimits where
Imports 
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.commuting-prisms-of-maps
open import foundation.commuting-squares-of-homotopies
open import foundation.commuting-squares-of-maps
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-extensionality
open import foundation.function-types
open import foundation.functoriality-dependent-pair-types
open import foundation.homotopies
open import foundation.retractions
open import foundation.retracts-of-types
open import foundation.sections
open import foundation.universe-levels
open import foundation.whiskering-homotopies-composition

open import synthetic-homotopy-theory.cocones-under-sequential-diagrams
open import synthetic-homotopy-theory.equivalences-sequential-diagrams
open import synthetic-homotopy-theory.morphisms-sequential-diagrams
open import synthetic-homotopy-theory.retracts-of-sequential-diagrams
open import synthetic-homotopy-theory.sequential-colimits
open import synthetic-homotopy-theory.sequential-diagrams
open import synthetic-homotopy-theory.universal-property-sequential-colimits

Idea

Given two sequential diagrams (A, a) and (B, b), their sequential colimits X and Y, and a morphism f : (A, a) → (B, b), there is a unique map g : X → Y, such that the diagram

        a₀      a₁      a₂
    A₀ ---> A₁ ---> A₂ ---> ⋯ ---> X
    |       |       |              |
 f₀ |       | f₁    | f₂           | g
    ∨       ∨       ∨              ∨
    B₀ ---> B₁ ---> B₂ ---> ⋯ ---> Y
        b₀      b₁      b₂

commutes.

The unique map corresponding to the identity morphism is the identity map id : X → X, and the unique map corresponding to a composite of two morphisms is the composite of the two unique maps for the individual morphisms.

In particular, when we also consider existence of the standard sequential colimits, we obtain a functorial action taking sequential diagrams and morphisms between them to their colimits and maps between them.

  (A, a)    A∞
    |       |
  f |   ↦   | f∞
    ∨       ∨
  (B, b)    B∞  .

Additionally, an equivalence of sequential diagrams induces an equivalence of their colimits.

Properties

A morphism of sequential diagrams induces a map of cocones

module _
  { l1 l2 l3 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  ( f : hom-sequential-diagram A B) {X : UU l3}
  where

  map-cocone-hom-sequential-diagram :
    cocone-sequential-diagram B X → cocone-sequential-diagram A X
  map-cocone-hom-sequential-diagram c =
    comp-hom-sequential-diagram A B (constant-sequential-diagram X) c f

A morphism of sequential diagrams induces a map of sequential colimits

The unique map g : X → Y such that the diagram

        a₀      a₁      a₂
    A₀ ---> A₁ ---> A₂ ---> ⋯ ---> X
    |       |       |              |
 f₀ |       | f₁    | f₂           | g
    ∨       ∨       ∨              ∨
    B₀ ---> B₁ ---> B₂ ---> ⋯ ---> Y
        b₀      b₁      b₂

commutes then induces a cocone under (A, a) with codomain Y, which is homotopic to the cocone under (B, b) precomposed by f.

This homotopy of cocones provides vertical commuting prisms of maps,

          Aₙ₊₁
         ∧ | \
       /   |   \
     /     |fₙ₊₁ ∨
    Aₙ ---------> X
    |      |      |
    |      ∨      |
 fₙ |     Bₙ₊₁    | g
    |    ∧   \    |
    |  /       \  |
    ∨/           ∨∨
    Bₙ ---------> Y ,

where the triangles are coherences of the cocones of the sequential colimits, the back left square is coherence of f, and the front and back right squares are provided by uniqueness of g.

module _
  { l1 l2 l3 l4 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  { X : UU l3} {c : cocone-sequential-diagram A X}
  ( up-c : universal-property-sequential-colimit c)
  { Y : UU l4} (c' : cocone-sequential-diagram B Y)
  ( f : hom-sequential-diagram A B)
  where

  map-sequential-colimit-hom-sequential-diagram : X → Y
  map-sequential-colimit-hom-sequential-diagram =
    map-universal-property-sequential-colimit up-c
      ( map-cocone-hom-sequential-diagram f c')

  htpy-cocone-map-sequential-colimit-hom-sequential-diagram :
    htpy-cocone-sequential-diagram
      ( cocone-map-sequential-diagram c
        ( map-sequential-colimit-hom-sequential-diagram))
      ( map-cocone-hom-sequential-diagram f c')
  htpy-cocone-map-sequential-colimit-hom-sequential-diagram =
    htpy-cocone-universal-property-sequential-colimit up-c
      ( map-cocone-hom-sequential-diagram f c')

  htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram :
    ( n : ℕ) →
    coherence-square-maps
      ( map-hom-sequential-diagram B f n)
      ( map-cocone-sequential-diagram c n)
      ( map-cocone-sequential-diagram c' n)
      ( map-sequential-colimit-hom-sequential-diagram)
  htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram =
    htpy-htpy-cocone-sequential-diagram
      ( htpy-cocone-map-sequential-colimit-hom-sequential-diagram)

  coherence-htpy-cocone-map-sequential-colimit-hom-sequential-diagram :
    ( n : ℕ) →
    coherence-square-homotopies
      ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram n)
      ( ( map-sequential-colimit-hom-sequential-diagram) ·l
        ( coherence-cocone-sequential-diagram c n))
      ( coherence-cocone-sequential-diagram
        ( map-cocone-hom-sequential-diagram f c')
        ( n))
      ( ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
          ( succ-ℕ n)) ·r
        ( map-sequential-diagram A n))
  coherence-htpy-cocone-map-sequential-colimit-hom-sequential-diagram =
    coherence-htpy-htpy-cocone-sequential-diagram
      htpy-cocone-map-sequential-colimit-hom-sequential-diagram

  prism-htpy-cocone-map-sequential-colimit-hom-sequential-diagram :
    ( n : ℕ) →
    vertical-coherence-prism-maps
      ( map-cocone-sequential-diagram c n)
      ( map-cocone-sequential-diagram c (succ-ℕ n))
      ( map-sequential-diagram A n)
      ( map-cocone-sequential-diagram c' n)
      ( map-cocone-sequential-diagram c' (succ-ℕ n))
      ( map-sequential-diagram B n)
      ( map-hom-sequential-diagram B f n)
      ( map-hom-sequential-diagram B f (succ-ℕ n))
      ( map-sequential-colimit-hom-sequential-diagram)
      ( coherence-cocone-sequential-diagram c n)
      ( inv-htpy
        ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram n))
      ( inv-htpy
        ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
          ( succ-ℕ n)))
      ( naturality-map-hom-sequential-diagram B f n)
      ( coherence-cocone-sequential-diagram c' n)
  prism-htpy-cocone-map-sequential-colimit-hom-sequential-diagram n =
    rotate-vertical-coherence-prism-maps
      ( map-cocone-sequential-diagram c n)
      ( map-cocone-sequential-diagram c (succ-ℕ n))
      ( map-sequential-diagram A n)
      ( map-cocone-sequential-diagram c' n)
      ( map-cocone-sequential-diagram c' (succ-ℕ n))
      ( map-sequential-diagram B n)
      ( map-hom-sequential-diagram B f n)
      ( map-hom-sequential-diagram B f (succ-ℕ n))
      ( map-sequential-colimit-hom-sequential-diagram)
      ( coherence-cocone-sequential-diagram c n)
      ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram n)
      ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
        ( succ-ℕ n))
      ( naturality-map-hom-sequential-diagram B f n)
      ( coherence-cocone-sequential-diagram c' n)
      ( inv-htpy
        ( coherence-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
          ( n)))

Homotopies between morphisms of sequential diagrams induce homotopies of corresponding maps between sequential colimits

module _
  { l1 l2 l3 l4 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  { X : UU l3} {c : cocone-sequential-diagram A X}
  ( up-c : universal-property-sequential-colimit c)
  { Y : UU l4} (c' : cocone-sequential-diagram B Y)
  { f g : hom-sequential-diagram A B} (H : htpy-hom-sequential-diagram B f g)
  where

  htpy-map-sequential-colimit-htpy-hom-sequential-diagram :
    map-sequential-colimit-hom-sequential-diagram up-c c' f ~
    map-sequential-colimit-hom-sequential-diagram up-c c' g
  htpy-map-sequential-colimit-htpy-hom-sequential-diagram =
    htpy-eq
      ( ap
        ( map-sequential-colimit-hom-sequential-diagram up-c c')
        ( eq-htpy-sequential-diagram _ _ f g H))

The identity morphism induces the identity map

module _
  { l1 l2 : Level} {A : sequential-diagram l1} {X : UU l2}
  { c : cocone-sequential-diagram A X}
  ( up-c : universal-property-sequential-colimit c)
  where

  htpy-preserves-id-map-sequential-colimit-hom-sequential-diagram :
    htpy-out-of-sequential-colimit c
      ( map-sequential-colimit-hom-sequential-diagram up-c c
        ( id-hom-sequential-diagram A))
      ( id)
  pr1 htpy-preserves-id-map-sequential-colimit-hom-sequential-diagram =
    htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c c
      ( id-hom-sequential-diagram A)
  pr2 htpy-preserves-id-map-sequential-colimit-hom-sequential-diagram n =
    ( coherence-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c c
      ( id-hom-sequential-diagram A)
      ( n)) ∙h
    ( ap-concat-htpy _
      ( ( right-unit-htpy) ∙h
        ( inv-htpy
          ( left-unit-law-left-whisker-comp
            ( coherence-cocone-sequential-diagram c n)))))

  preserves-id-map-sequential-colimit-hom-sequential-diagram :
    ( map-sequential-colimit-hom-sequential-diagram up-c c
      ( id-hom-sequential-diagram A)) ~
    ( id)
  preserves-id-map-sequential-colimit-hom-sequential-diagram =
    htpy-htpy-out-of-sequential-colimit up-c
      ( htpy-preserves-id-map-sequential-colimit-hom-sequential-diagram)

Composition of morphisms induces composition of functions

module _
  { l1 l2 l3 l4 l5 l6 : Level}
  { A : sequential-diagram l1} {B : sequential-diagram l2}
  { C : sequential-diagram l3}
  { X : UU l4} {c : cocone-sequential-diagram A X}
  ( up-c : universal-property-sequential-colimit c)
  { Y : UU l5} {c' : cocone-sequential-diagram B Y}
  ( up-c' : universal-property-sequential-colimit c')
  { Z : UU l6} (c'' : cocone-sequential-diagram C Z)
  ( g : hom-sequential-diagram B C) (f : hom-sequential-diagram A B)
  where

  htpy-preserves-comp-map-sequential-colimit-hom-sequential-diagram :
    htpy-out-of-sequential-colimit c
      ( map-sequential-colimit-hom-sequential-diagram up-c c''
        ( comp-hom-sequential-diagram A B C g f))
      ( ( map-sequential-colimit-hom-sequential-diagram up-c' c'' g) ∘
        ( map-sequential-colimit-hom-sequential-diagram up-c c' f))
  pr1 htpy-preserves-comp-map-sequential-colimit-hom-sequential-diagram n =
    ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c c''
      ( comp-hom-sequential-diagram A B C g f)
      ( n)) ∙h
    ( pasting-vertical-coherence-square-maps
      ( map-cocone-sequential-diagram c n)
      ( map-hom-sequential-diagram B f n)
      ( map-sequential-colimit-hom-sequential-diagram up-c c' f)
      ( map-cocone-sequential-diagram c' n)
      ( map-hom-sequential-diagram C g n)
      ( map-sequential-colimit-hom-sequential-diagram up-c' c'' g)
      ( map-cocone-sequential-diagram c'' n)
      ( inv-htpy
        ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c c'
          ( f)
          ( n)))
      ( inv-htpy
        ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c'
          ( c'')
          ( g)
          ( n))))
  pr2 htpy-preserves-comp-map-sequential-colimit-hom-sequential-diagram n =
    ( right-whisker-concat-coherence-square-homotopies
      ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c c''
        ( comp-hom-sequential-diagram A B C g f)
        ( n))
      ( ( map-sequential-colimit-hom-sequential-diagram up-c c''
          ( comp-hom-sequential-diagram A B C g f)) ·l
        ( coherence-cocone-sequential-diagram c n))
      ( coherence-cocone-sequential-diagram
        ( map-cocone-hom-sequential-diagram
          ( comp-hom-sequential-diagram A B C g f)
          ( c''))
        ( n))
      ( ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
          ( up-c)
          ( c'')
          ( comp-hom-sequential-diagram A B C g f)
          ( succ-ℕ n)) ·r
        ( map-sequential-diagram A n))
      ( coherence-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c
        ( c'')
        ( comp-hom-sequential-diagram A B C g f)
        ( n))
      ( _)) ∙h
    ( ap-concat-htpy
      ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c c''
        ( comp-hom-sequential-diagram A B C g f)
        ( n))
      ( ( assoc-htpy
          ( ( coherence-cocone-sequential-diagram c'' n) ·r
            ( ( map-hom-sequential-diagram C g n) ∘
              ( map-hom-sequential-diagram B f n)))
          ( map-cocone-sequential-diagram c'' (succ-ℕ n) ·l _)
          ( _)) ∙h
        ( pasting-vertical-coherence-prism-maps
          ( map-cocone-sequential-diagram c n)
          ( map-cocone-sequential-diagram c (succ-ℕ n))
          ( map-sequential-diagram A n)
          ( map-cocone-sequential-diagram c' n)
          ( map-cocone-sequential-diagram c' (succ-ℕ n))
          ( map-sequential-diagram B n)
          ( map-hom-sequential-diagram B f n)
          ( map-hom-sequential-diagram B f (succ-ℕ n))
          ( map-sequential-colimit-hom-sequential-diagram up-c c' f)
          ( map-cocone-sequential-diagram c'' n)
          ( map-cocone-sequential-diagram c'' (succ-ℕ n))
          ( map-sequential-diagram C n)
          ( map-hom-sequential-diagram C g n)
          ( map-hom-sequential-diagram C g (succ-ℕ n))
          ( map-sequential-colimit-hom-sequential-diagram up-c' c'' g)
          ( coherence-cocone-sequential-diagram c n)
          ( inv-htpy
            ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
              ( up-c)
              ( c')
              ( f)
              ( n)))
          ( inv-htpy
            ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
              ( up-c)
              ( c')
              ( f)
              ( succ-ℕ n)))
          ( naturality-map-hom-sequential-diagram B f n)
          ( coherence-cocone-sequential-diagram c' n)
          ( inv-htpy
            ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
              ( up-c')
              ( c'')
              ( g)
              ( n)))
          ( inv-htpy
            ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
              ( up-c')
              ( c'')
              ( g)
              ( succ-ℕ n)))
          ( naturality-map-hom-sequential-diagram C g n)
          ( coherence-cocone-sequential-diagram c'' n)
          ( prism-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c
            ( c')
            ( f)
            ( n))
          ( prism-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
            ( up-c')
            ( c'')
            ( g)
            ( n))))) ∙h
    ( inv-htpy-assoc-htpy
      ( htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram up-c c''
        ( comp-hom-sequential-diagram A B C g f)
        ( n))
      ( _)
      ( ( ( map-sequential-colimit-hom-sequential-diagram up-c' c'' g) ∘
          ( map-sequential-colimit-hom-sequential-diagram up-c c' f)) ·l
        ( coherence-cocone-sequential-diagram c n)))

  preserves-comp-map-sequential-colimit-hom-sequential-diagram :
    ( map-sequential-colimit-hom-sequential-diagram up-c c''
      ( comp-hom-sequential-diagram A B C g f)) ~
    ( ( map-sequential-colimit-hom-sequential-diagram up-c' c'' g) ∘
      ( map-sequential-colimit-hom-sequential-diagram up-c c' f))
  preserves-comp-map-sequential-colimit-hom-sequential-diagram =
    htpy-htpy-out-of-sequential-colimit up-c
      ( htpy-preserves-comp-map-sequential-colimit-hom-sequential-diagram)

A retract of sequential diagrams induces a retract of sequential colimits

Additionally, the underlying map of the retraction is strictly equal to the map induced by the retraction of sequential diagrams.

module _
  { l1 l2 l3 l4 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  { X : UU l3} {c : cocone-sequential-diagram A X}
  ( up-c : universal-property-sequential-colimit c)
  { Y : UU l4} {c' : cocone-sequential-diagram B Y}
  ( up-c' : universal-property-sequential-colimit c')
  ( R : A retract-of-sequential-diagram B)
  where

  map-inclusion-sequential-colimit-retract-sequential-diagram : X → Y
  map-inclusion-sequential-colimit-retract-sequential-diagram =
    map-sequential-colimit-hom-sequential-diagram up-c c'
      ( inclusion-retract-sequential-diagram A B R)

  map-sequential-colimit-retract-sequential-diagram : Y → X
  map-sequential-colimit-retract-sequential-diagram =
    map-sequential-colimit-hom-sequential-diagram up-c' c
      ( hom-retraction-retract-sequential-diagram A B R)

  abstract
    is-retraction-map-sequential-colimit-retract-sequential-diagram :
      is-retraction
        ( map-inclusion-sequential-colimit-retract-sequential-diagram)
        ( map-sequential-colimit-retract-sequential-diagram)
    is-retraction-map-sequential-colimit-retract-sequential-diagram =
      ( inv-htpy
        ( preserves-comp-map-sequential-colimit-hom-sequential-diagram up-c
          ( up-c')
          ( c)
          ( hom-retraction-retract-sequential-diagram A B R)
          ( inclusion-retract-sequential-diagram A B R))) ∙h
      ( htpy-map-sequential-colimit-htpy-hom-sequential-diagram up-c c
        ( is-retraction-hom-retraction-retract-sequential-diagram A B R)) ∙h
      ( preserves-id-map-sequential-colimit-hom-sequential-diagram up-c)

  retraction-sequential-colimit-retract-sequential-diagram :
    retraction map-inclusion-sequential-colimit-retract-sequential-diagram
  pr1 retraction-sequential-colimit-retract-sequential-diagram =
    map-sequential-colimit-retract-sequential-diagram
  pr2 retraction-sequential-colimit-retract-sequential-diagram =
    is-retraction-map-sequential-colimit-retract-sequential-diagram

  retract-sequential-colimit-retract-sequential-diagram :
    X retract-of Y
  pr1 retract-sequential-colimit-retract-sequential-diagram =
    map-inclusion-sequential-colimit-retract-sequential-diagram
  pr2 retract-sequential-colimit-retract-sequential-diagram =
    retraction-sequential-colimit-retract-sequential-diagram

An equivalence of sequential diagrams induces an equivalence of sequential colimits

Additionally, the underlying map of the inverse equivalence is definitionally equal to the map induced by the inverse of the equivalence of sequential diagrams, i.e. it is the unique map g : Y → X making the diagram

           b₀      b₁      b₂
       B₀ ---> B₁ ---> B₂ ---> ⋯ ---> Y
       |       |       |              |
  e₀⁻¹ |       | e₁⁻¹  | e₂⁻¹         | g
       ∨       ∨       ∨              ∨
       A₀ ---> A₁ ---> A₂ ---> ⋯ ---> X
           a₀      a₁      a₂

commute.

module _
  { l1 l2 l3 l4 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  { X : UU l3} {c : cocone-sequential-diagram A X}
  ( up-c : universal-property-sequential-colimit c)
  { Y : UU l4} {c' : cocone-sequential-diagram B Y}
  ( up-c' : universal-property-sequential-colimit c')
  ( e : equiv-sequential-diagram A B)
  where

  map-equiv-sequential-colimit-equiv-sequential-diagram : X → Y
  map-equiv-sequential-colimit-equiv-sequential-diagram =
    map-sequential-colimit-hom-sequential-diagram up-c c'
      ( hom-equiv-sequential-diagram B e)

  map-inv-equiv-sequential-colimit-equiv-sequential-diagram : Y → X
  map-inv-equiv-sequential-colimit-equiv-sequential-diagram =
    map-sequential-colimit-hom-sequential-diagram up-c' c
      ( hom-inv-equiv-sequential-diagram B e)

  abstract
    is-section-map-inv-equiv-sequential-colimit-equiv-sequential-diagram :
      is-section
        ( map-equiv-sequential-colimit-equiv-sequential-diagram)
        ( map-inv-equiv-sequential-colimit-equiv-sequential-diagram)
    is-section-map-inv-equiv-sequential-colimit-equiv-sequential-diagram =
      ( inv-htpy
        ( preserves-comp-map-sequential-colimit-hom-sequential-diagram up-c'
          ( up-c)
          ( c')
          ( hom-equiv-sequential-diagram B e)
          ( hom-inv-equiv-sequential-diagram B e))) ∙h
      ( htpy-map-sequential-colimit-htpy-hom-sequential-diagram up-c' c'
        ( is-section-inv-equiv-sequential-diagram _ e)) ∙h
      ( preserves-id-map-sequential-colimit-hom-sequential-diagram up-c')

    is-retraction-map-inv-equiv-sequential-colimit-equiv-sequential-diagram :
      is-retraction
        ( map-equiv-sequential-colimit-equiv-sequential-diagram)
        ( map-inv-equiv-sequential-colimit-equiv-sequential-diagram)
    is-retraction-map-inv-equiv-sequential-colimit-equiv-sequential-diagram =
      is-retraction-map-sequential-colimit-retract-sequential-diagram up-c up-c'
        ( retract-equiv-sequential-diagram e)

  is-equiv-map-equiv-sequential-colimit-equiv-sequential-diagram :
    is-equiv map-equiv-sequential-colimit-equiv-sequential-diagram
  is-equiv-map-equiv-sequential-colimit-equiv-sequential-diagram =
    is-equiv-is-invertible
      ( map-inv-equiv-sequential-colimit-equiv-sequential-diagram)
      ( is-section-map-inv-equiv-sequential-colimit-equiv-sequential-diagram)
      ( is-retraction-map-inv-equiv-sequential-colimit-equiv-sequential-diagram)

  equiv-sequential-colimit-equiv-sequential-diagram : X ≃ Y
  pr1 equiv-sequential-colimit-equiv-sequential-diagram =
    map-equiv-sequential-colimit-equiv-sequential-diagram
  pr2 equiv-sequential-colimit-equiv-sequential-diagram =
    is-equiv-map-equiv-sequential-colimit-equiv-sequential-diagram

Functoriality of taking the standard sequential colimit

All of the above specializes to the case where X is the standard sequential colimit A∞ and Y is the standard sequential colimit B∞. In that case, we denote the unique map g corresponding to a morphism of diagrams f by f∞ : A∞ → B∞.

A morphism of sequential diagrams induces a map of standard sequential colimits

module _
  { l1 l2 : Level} {A : sequential-diagram l1} (B : sequential-diagram l2)
  ( f : hom-sequential-diagram A B)
  where

  map-hom-standard-sequential-colimit :
    standard-sequential-colimit A → standard-sequential-colimit B
  map-hom-standard-sequential-colimit =
    map-sequential-colimit-hom-sequential-diagram
      ( up-standard-sequential-colimit)
      ( cocone-standard-sequential-colimit B)
      ( f)

  htpy-cocone-map-hom-standard-sequential-colimit :
    htpy-cocone-sequential-diagram
      ( cocone-map-sequential-diagram
        ( cocone-standard-sequential-colimit A)
        ( map-hom-standard-sequential-colimit))
      ( map-cocone-hom-sequential-diagram f
        ( cocone-standard-sequential-colimit B))
  htpy-cocone-map-hom-standard-sequential-colimit =
    htpy-cocone-map-sequential-colimit-hom-sequential-diagram
      ( up-standard-sequential-colimit)
      ( cocone-standard-sequential-colimit B)
      ( f)

  htpy-htpy-cocone-map-hom-standard-sequential-colimit :
    ( n : ℕ) →
    coherence-square-maps
      ( map-hom-sequential-diagram B f n)
      ( map-cocone-standard-sequential-colimit n)
      ( map-cocone-standard-sequential-colimit n)
      ( map-hom-standard-sequential-colimit)
  htpy-htpy-cocone-map-hom-standard-sequential-colimit =
    htpy-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
      ( up-standard-sequential-colimit)
      ( cocone-standard-sequential-colimit B)
      ( f)

  coherence-htpy-cocone-map-hom-standard-sequential-colimit :
    ( n : ℕ) →
    coherence-square-homotopies
      ( htpy-htpy-cocone-map-hom-standard-sequential-colimit n)
      ( ( map-hom-standard-sequential-colimit) ·l
        ( coherence-cocone-standard-sequential-colimit n))
      ( coherence-cocone-sequential-diagram
          ( map-cocone-hom-sequential-diagram f
            ( cocone-standard-sequential-colimit B))
          ( n))
      ( ( htpy-htpy-cocone-map-hom-standard-sequential-colimit (succ-ℕ n)) ·r
        ( map-sequential-diagram A n))
  coherence-htpy-cocone-map-hom-standard-sequential-colimit =
    coherence-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
      ( up-standard-sequential-colimit)
      ( cocone-standard-sequential-colimit B)
      ( f)

  prism-htpy-cocone-map-hom-standard-sequential-colimit :
    ( n : ℕ) →
    vertical-coherence-prism-maps
      ( map-cocone-standard-sequential-colimit n)
      ( map-cocone-standard-sequential-colimit (succ-ℕ n))
      ( map-sequential-diagram A n)
      ( map-cocone-standard-sequential-colimit n)
      ( map-cocone-standard-sequential-colimit (succ-ℕ n))
      ( map-sequential-diagram B n)
      ( map-hom-sequential-diagram B f n)
      ( map-hom-sequential-diagram B f (succ-ℕ n))
      ( map-hom-standard-sequential-colimit)
      ( coherence-cocone-standard-sequential-colimit n)
      ( inv-htpy (htpy-htpy-cocone-map-hom-standard-sequential-colimit n))
      ( inv-htpy
        ( htpy-htpy-cocone-map-hom-standard-sequential-colimit (succ-ℕ n)))
      ( naturality-map-hom-sequential-diagram B f n)
      ( coherence-cocone-standard-sequential-colimit n)
  prism-htpy-cocone-map-hom-standard-sequential-colimit =
    prism-htpy-cocone-map-sequential-colimit-hom-sequential-diagram
      ( up-standard-sequential-colimit)
      ( cocone-standard-sequential-colimit B)
      ( f)

Homotopies between morphisms of sequential diagrams induce homotopies of maps of standard sequential colimits

module _
  { l1 l2 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  { f g : hom-sequential-diagram A B} (H : htpy-hom-sequential-diagram B f g)
  where

  htpy-map-hom-standard-sequential-colimit-htpy-hom-sequential-diagram :
    map-hom-standard-sequential-colimit B f ~
    map-hom-standard-sequential-colimit B g
  htpy-map-hom-standard-sequential-colimit-htpy-hom-sequential-diagram =
    htpy-map-sequential-colimit-htpy-hom-sequential-diagram
      ( up-standard-sequential-colimit)
      ( cocone-standard-sequential-colimit B)
      ( H)

The identity morphism induces the identity map on the standard sequential colimit

We have id∞ ~ id : A∞ → A∞.

module _
  { l1 : Level} {A : sequential-diagram l1}
  where

  preserves-id-map-hom-standard-sequential-colimit :
    ( map-hom-standard-sequential-colimit A
      ( id-hom-sequential-diagram A)) ~
    ( id)
  preserves-id-map-hom-standard-sequential-colimit =
    preserves-id-map-sequential-colimit-hom-sequential-diagram
      ( up-standard-sequential-colimit)

Forming standard sequential colimits preserves composition of morphisms of sequential diagrams

We have (f ∘ g)∞ ~ (f∞ ∘ g∞) : A∞ → C∞.

module _
  { l1 l2 l3 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  { C : sequential-diagram l3}
  ( g : hom-sequential-diagram B C) (f : hom-sequential-diagram A B)
  where

  preserves-comp-map-hom-standard-sequential-colimit :
    ( map-hom-standard-sequential-colimit C
      ( comp-hom-sequential-diagram A B C g f)) ~
    ( ( map-hom-standard-sequential-colimit C g) ∘
      ( map-hom-standard-sequential-colimit B f))
  preserves-comp-map-hom-standard-sequential-colimit =
    preserves-comp-map-sequential-colimit-hom-sequential-diagram
      ( up-standard-sequential-colimit)
      ( up-standard-sequential-colimit)
      ( cocone-standard-sequential-colimit C)
      ( g)
      ( f)

An equivalence of sequential diagrams induces an equivalence of standard sequential colimits

Additionally, the underlying map of the inverse equivalence is definitionally equal to the map induced by the inverse of the equivalence of sequential diagrams, i.e. (e∞)⁻¹ = (e⁻¹)∞.

module _
  { l1 l2 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  ( e : equiv-sequential-diagram A B)
  where

  map-equiv-standard-sequential-colimit :
    standard-sequential-colimit A → standard-sequential-colimit B
  map-equiv-standard-sequential-colimit =
    map-equiv-sequential-colimit-equiv-sequential-diagram
      ( up-standard-sequential-colimit)
      ( up-standard-sequential-colimit)
      ( e)

  map-inv-equiv-standard-sequential-colimit :
    standard-sequential-colimit B → standard-sequential-colimit A
  map-inv-equiv-standard-sequential-colimit =
    map-inv-equiv-sequential-colimit-equiv-sequential-diagram
      ( up-standard-sequential-colimit)
      ( up-standard-sequential-colimit)
      ( e)

  equiv-equiv-standard-sequential-colimit :
    standard-sequential-colimit A ≃ standard-sequential-colimit B
  equiv-equiv-standard-sequential-colimit =
    equiv-sequential-colimit-equiv-sequential-diagram
      ( up-standard-sequential-colimit)
      ( up-standard-sequential-colimit)
      ( e)

A retract of sequential diagrams induces a retract of standard sequential colimits

Additionally, the underlying map of the retraction is definitionally equal to the map induced by the retraction of sequential diagrams.

module _
  { l1 l2 : Level} {A : sequential-diagram l1} {B : sequential-diagram l2}
  ( R : A retract-of-sequential-diagram B)
  where

  map-inclusion-retract-standard-sequential-colimit :
    standard-sequential-colimit A → standard-sequential-colimit B
  map-inclusion-retract-standard-sequential-colimit =
    map-hom-standard-sequential-colimit B
      ( inclusion-retract-sequential-diagram A B R)

  map-hom-retraction-retract-standard-sequential-colimit :
    standard-sequential-colimit B → standard-sequential-colimit A
  map-hom-retraction-retract-standard-sequential-colimit =
    map-hom-standard-sequential-colimit A
      ( hom-retraction-retract-sequential-diagram A B R)

  retract-retract-standard-sequential-colimit :
    (standard-sequential-colimit A) retract-of (standard-sequential-colimit B)
  retract-retract-standard-sequential-colimit =
    retract-sequential-colimit-retract-sequential-diagram
      ( up-standard-sequential-colimit)
      ( up-standard-sequential-colimit)
      ( R)

References

[SDR20]
Kristina Sojakova, Floris van Doorn, and Egbert Rijke. Sequential Colimits in Homotopy Type Theory. In Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '20, 845–858. Association for Computing Machinery, 07 2020. doi:10.1145/3373718.3394801.

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