Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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.
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
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
This is the homotopy equalizer of $f$ and $g$.
See also
and
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.
In simplicial sets with simplicial weak equivalences, the homotopy equalizer of $f,g\colon A\to B$ can be computed as the pullback
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
where
Last revised on June 26, 2022 at 09:53:06. See the history of this page for a list of all contributions to it.