nLab homotopy equalizer

Idea

Homotopy equalizers are a special case of homotopy limits, when the indexing diagram is the walking parallel pair, which consists 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 equalizers 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.

In the absence of nontrivial homotopies (in a bare 1-category), homotopy equalizers reduce to ordinary equalizers.

Properties

Relation to homotopy pullbacks

In any model category, the homotopy equalizer of a pair of arrows $f,g\colon A\to B$ can be computed as follows.

First, if $B$ is not fibrant and the model category is not right proper, construct a fibrant replacement $r\colon B\to R B$ and replace $(f,g)$ with $(r f,r g)$.

Assume now that $B$ is fibrant or the model category is right proper.

In the special case when the map

$$(f,g) \;\colon\; A \longrightarrow B\times B$$

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

In the general case, factor the diagonal map $\Delta \;\colon\; B\to B\times B$ as a weak equivalence $B \to P B$ followed by a fibration $P B \longrightarrow B\times B$, then compute the (ordinary) pullback of

$$A \longrightarrow B\times B \longleftarrow P B \,.$$

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

See also

• this Prop. at homotopy pullback

and

• this Prop at Mayer-Vietoris sequence.

Relation to homotopy fibers

For pointed model categories, the homotopy equalizer of $f\colon A\to B$ and the zero morphism $0\colon A\to B$ is known as the homotopy fiber of $f$. See there for more information.

Examples

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

$$A\times_{B\times B}B^{\Delta^1}$$

if $B$ is a Kan complex. (Otherwise, compose with a fibrant replacement like $B\to Ex^\infty B$ first.) The same formula works for topological spaces with weak homotopy equivalences, using $\Delta=[0,1]$.

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

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

where

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

Related notions

• homotopy coequalizer

• product

• homotopy limit

• homotopy pullback

• homotopy product

• homotopy totalization