nLab
homotopy equalizer

Contents

Context

Limits and colimits

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

Contents

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 010\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.

Computation

In any model category, the homotopy equalizer of a pair of arrows f,g:ABf,g\colon A\to B can be computed as follows. First, if BB is not fibrant and the model category is not right proper, construct a fibrant replacement r:BRBr\colon B\to R B and replace (f,g)(f,g) with (rf,rg)(r f,r g).

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

In the special case when the map

(f,g):AB×B(f,g)\colon A\to B\times B

happens to be a fibration, we can compute the ordinary equalizer of ff and gg, which is a homotopy equalizer.

In the general case, factor the diagonal map Δ:BB×B\Delta\colon B\to B\times B as a weak equivalence BPBB \to P B followed by a fibration PBB×BP B\to B\times B, then compute the (ordinary) pullback of

AB×BPB.A\to B\times B\leftarrow P B.

This is the homotopy equalizer of ff and gg.

Homotopy fibers

For pointed model categories, the homotopy equalizer of f:ABf\colon A\to B and the zero morphism 0:AB0\colon A\to B is known as the homotopy fiber of ff. See there for more information.

Examples

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

A× B×BB Δ 1A\times_{B\times B}B^{\Delta^1}

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

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

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

where

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

Last revised on January 31, 2021 at 16:57:52. See the history of this page for a list of all contributions to it.