nLab homotopy coequalizer

Idea

In generalization of the notion of equalizers from category theory to homotopy theory, homotopy coequalizers are the special case of homotopy colimits, where 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 a homotopy pushout as follows (compare the ordinary expression of coequalizer as pushouts, here).

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

$$(f,g) \,\colon\, A\sqcup A \longrightarrow B$$

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

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

$$CA \longleftarrow A\sqcup A \overset{(f,g)}{\longrightarrow} B \,.$$

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

$$(\Delta^1 \times A) \overset{A\sqcup A}{\amalg} B \,.$$

The analogous formula works for topological spaces with weak homotopy equivalences, then using the closed interval $\Delta=[0,1]$ and also known as the double mapping cylinder-construction, see for instance Dwyer, Farjoun & Ravenel (1999), p. 1856 (cf. mapping cylinder).

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

$$B \oplus A[1] \,,$$

where

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

Related notions

• mapping cylinder

• coproduct

• homotopy colimit

• homotopy pushout

• homotopy coproduct

• homotopy realization

• homotopy equalizer

References

• William G. Dwyer, Emmanuel Dror Farjoun, Douglas C. Ravenel, pp. 1856 in: Bousfield Localizations of Classifying Spaces of Nilpotent Groups, Proceedings of the American Mathematical Society 127 6 (1999) [jstor:119499]

• Topospaces, Double mapping cylinder