A weak homotopy equivalence is a map between topological spaces or simplicial sets or similar which induces isomorphisms on all homotopy groups. In homological algebra this is called a quasi-isomorphism.
The localization or simplicial localization of the categories Top and sSet at the weak homotopy equivalences used as weak equivalences yields the standard homotopy category Ho(Top) and Ho(sSet) or the (∞,1)-category of ∞-groupoids/homotopy types, respectively.
Weak homotopy equivalences are named after homotopy equivalences. They can be identified with homotopy equivalences after one allows to replace the domains by a resolution. The corresponding notions in homological algebra are quasi-isomorphisms and chain homotopy-equivalences.
From another perspective, the notion of weak homotopy equivalence is ‘observational’, in that a map is a weak homotopy equivalence if when we look at it through the observations that we can make of it using homotopy groups or even the fundamental infinity-groupoid, it looks like an equivalence. In contrast, homotopy equivalence is more ‘constructive’; in that is a homotopy equivalence if there exists an inverse for it (up to homotopy, of course). Note that both of these notions are weaker than mere isomorphism of topological spaces (homeomorphism) and so can be considered examples of weak equivalences.
There are actually two related concepts here: whether two spaces are weakly homotopy equivalent and whether a map between spaces is a weak homotopy equivalence. The former is usually defined in terms of the latter.
Accordingly, weak homotopy equivalences are the weak equivalences in the standard Quillen model structure on topological spaces and the Quillen model structure on simplicial sets, and also in the mixed model structure.
A continuous map is a weak homotopy equivalence precisely if for all and for all commuting diagrams of continuous maps of the form
where the left morphism is the inclusion of the -sphere as the boundary of the -ball, there exists a continuous map that makes the resulting upper triangle commute and such that the lower triangle commutes up to a homotopy
which is constant along .
In this form the statement and its proof appears in (Jardine) (where it is also generalized to weak equivalences in a model structure on simplicial presheaves). See also around (Lurie, prop. 22.214.171.124). The relevant arguments are spelled out in (May, section 9.6). A variant is called the HELP lemma in (Vogt).
Every homotopy equivalence is a weak homotopy equivalence.
It requires a little bit of thought to prove this, because and its homotopy inverse need not preserve any chosen basepoint. But for any and any , we have a diagram
We discuss the equivalence relation generated by weak homotopy equivalence, called (weak) homotopy type. For the “abelianized” analog of this situation see at quasi-isomorphism the section Relation to homology type.
Reflexivity and transitivity are trivially checked. A counterexample to symmetry is example 1 below.
But we can consider the genuine equivalence relation generated by weak homotopy equivalence:
Equivalently this means that and have the same (weak) homotopy type if there exists a zigzag of weak homotopy equivalences
Two spaces and may have isomorphic homotopy groups without being weak homotopy equivalence: for this all the isomorphisms must be induced by an actual map , as in the above definition.
However, if, roughly, one remembers, how all the homotopy groups act on each other, then this is enough information to exhibit the full homotopy type. This collection of data is called the Postnikov tower decomposition of a homotopy type.
A map of simplicial sets is called a weak homotopy equivalence equivalently if its geometric realization is a weak homotopy equivalence of topological spaces, as above. (Since the geometric realization of any simpicial set is a CW complex, in this case its geometric realization is actually a homotopy equivalence.)
Likewise, a functor between small categories is sometimes said to be a weak homotopy equivalence if its nerve is a weak homotopy equivalence of simplicial sets, hence of topological spaces after geometric realization of categories. These are the weak equivalences in the Thomason model structure on categories (not the canonical model structure). The statement of Quillen's theorem A and Quillen's theorem B in in this contex.
Note that in some of these cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of “homotopy equivalence” from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map with an inverse and simplicial homotopies and relating and to identities.
A different direction of generalization is the notion of a homotopy equivalence of toposes.
We discuss examples of weak homotopy equivalences that have no weak homotopy equivalence going the other way, according to prop. 4 above.
There is a continuous function which is a weak homotopy equivalence, hence in particular . But every continuous map the other way round has to induce the trivial map on .
This is the simplest in a class of counter-examples discussed in (McCord).
homotopy equivalence, weak homotopy equivalence
A general account is for instance in section 9.6 of
The characterization of weak homotopy equivalences by lifts up to homotopy seems is in
For related and general discussion see also section 6.5 of
Examples for the non-symmetry of the weak homotopy equivalence relation are in