Skip to content

Commit

Permalink
Put specialization to standard sequential colimits into its heading
Browse files Browse the repository at this point in the history
  • Loading branch information
VojtechStep committed Jan 2, 2024
1 parent 2607f0a commit 148c592
Showing 1 changed file with 20 additions and 7 deletions.
Original file line number Diff line number Diff line change
Expand Up @@ -63,7 +63,15 @@ composition of morphisms is taken to composition of functions,

A corollary of these facts is that taking the
[standard sequential colimit](synthetic-homotopy-theory.sequential-colimits.md)
is a functorial action `(A, a) ↦ A∞`.
is a functorial action

```text
(A, a) A∞
| |
f | ↦ | f∞
V V
(B, b) B∞ .
```

Additionally, an
[equivalence of sequential diagrams](synthetic-homotopy-theory.equivalences-sequential-diagrams.md)
Expand Down Expand Up @@ -535,7 +543,12 @@ module _
is-equiv-map-equiv-sequential-colimit-equiv-sequential-diagram
```

### A morphism of sequential diagrams induces a map of standard sequential colimits
### 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∞`.

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

```agda
module _
Expand Down Expand Up @@ -620,7 +633,7 @@ module _
( f)
```

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

```agda
module _
Expand All @@ -638,7 +651,7 @@ module _
( H)
```

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

We have `id∞ ~ id : A∞ A∞`.

Expand All @@ -656,7 +669,7 @@ module _
( up-standard-sequential-colimit)
```

### Forming standard sequential colimits preserves composition of morphisms of sequential diagrams
#### Forming standard sequential colimits preserves composition of morphisms of sequential diagrams

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

Expand All @@ -681,7 +694,7 @@ module _
( f)
```

### An equivalence of sequential diagrams induces an equivalence of standard sequential colimits
#### 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
Expand Down Expand Up @@ -718,7 +731,7 @@ module _
( e)
```

### A retract of sequential diagrams induces a retract of standard sequential colimits
#### 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.
Expand Down

0 comments on commit 148c592

Please sign in to comment.