# nLab homotopy coequalizer

Contents

### Context

#### Limits and colimits

limits and colimits

## (∞,1)-Categorical

### Model-categorical

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Idea

Homotopy coequalizers are a special case of homotopy colimits, when the indexing diagram is the walking parallel pair, consisting of a pair of parallel morphisms, i.e., two objects, 0 and 1, and exactly two nonidentity morphisms, both of the form $0\to 1$.

Homotopy coequalizers can be defined in any relative category, just like homotopy colimits, but practical computations are typically carried out in presence of additional structures such as model structures.

## Computation

In any model category, the homotopy coequalizer of a pair of arrows $f,g\colon A\to B$ can be computed as follows. First, if $A$ is not cofibrant and the model category is not left proper, construct a cofibrant replacement $q\colon QA\to A$ and replace $(f,g)$ with $(f q,g q)$.

Assume now that $A$ is cofibrant or the model category is left proper.

In the special case when the map

$[f,g]\colon A\sqcup A\to B$

happens to be a cofibration, we can compute the ordinary coequalizer of $f$ and $g$, which is a homotopy coequalizer.

In the general case, factor the codiagonal map $\nabla\colon A\sqcup A\to A$ as a cofibration $A\sqcup A\to C A$ followed by a weak equivalence $CA\to A$, then compute the (ordinary) pushout of

$CA\leftarrow A\sqcup A \to B.$

This is the homotopy coequalizer of $f$ and $g$.

## Examples

In simplicial sets with simplicial weak equivalences, the homotopy coequalizer of $f,g\colon A\to B$ can be computed as the pushout

$\Delta^1\times A\sqcup_{A\sqcup A} B.$

The same formula works for topological spaces with weak homotopy equivalences, using $\Delta=[0,1]$.

For chain complexes with quasi-isomorphisms, the homotopy coequalizer can be computed (expanding the analogous formula with $C A =\mathrm{N} \mathbf{Z} [\Delta^1]\otimes A$) as

$B\oplus A,$

where

$d(b\oplus a)=d b+f(a)-g(a)\oplus d a.$

Last revised on January 31, 2021 at 16:58:36. See the history of this page for a list of all contributions to it.