related by the Dold-Kan correspondence
In the category of ‘spaces’, by ‘invariants’ we often mean ‘homotopy invariants’, so as well as giving a functor on the category of spaces taking values, say, in the category of Abelian groups, such an invariant also induces one on the ‘homotopy category’, that is the category of spaces and homotopy classes of maps between them. This ‘homotopy category’ construction can be viewed as a general construction on categories having a congruence relation on their hom-sets, and hence as a sort of way of extracting an interesting and hopefully more tractable, category from a ‘higher category’ of some sort, perhaps a 2-category or more generally an -category. The relationship between the higher category, the basic category, say of spaces, and this ‘homotopy category’ is simple, but needs looking at from the nPOV.
Quite often one encounters an ordinary category which is known in some way or other to be the -categorical truncation of a higher category . Standard examples include the categories SimpSet of simplicial sets (or Top of topological spaces) and of chain complexes of abelian groups. Both are obtained from full (infinity,1)-categories by forgetting higher morphisms.
The most important information that is lost by forgetting the higher morphisms of a higher category is that about which 1-morphisms are, while not isomorphisms, invertible up to higher cells, i.e. equivalences.
To the full -category is canonically associated a 1-category called the homotopy category of an (infinity,1)-category, which is obtained from not by simply forgetting the higher morphisms, but by quotienting them out, i.e. by remembering the equivalence classes of 1-morphisms. In the -category Top (restricted to sufficiently nice objects, such as compactly generated weakly Hausdorff topological spaces) these higher morphisms are literally the homotopies between 1-morphisms, and more generally one tends to address higher cells in -categories as homotopies. Therefore the name homotopy category of an -category for . In particular is the standard homotopy category originally introduced in topology.
Now a slightly different viewpoint comes in that interacts neatly with this one of ‘dividing out by 1-morphisms’. Suppose we are given just the truncated 1-category , but now equipped with the structure of a category with weak equivalences which indicates which morphisms in are to be regarded as equivalences in a higher categorical context, there is a universal solution to the problem of finding a category equipped with a functor such that sends all (morphisms labeled as) weak equivalences in to isomorphisms in .
In good situations, one may also find an -category corresponding to , and the notions of homotopy category and then coincide.
This is, in particular, the case when is equipped with the structure of a combinatorial simplicial model category and is the -category presented by with its model structure. (For instance HTT, remark A.3.1.8).
We thus have several interrelated notions of homotopy category, which in most useful contexts more or less coincide. Because of that, the term tends to be used widely, and the context then determines the exact definition to apply. We will give some of the most common ones, selected for their use in other entries.
Given a simplicially enriched category , we can form for each pair of objects, , of objects of , the set, , of connected components of the ‘function space’ . As preserves finite limits, this gives a category, denoted . As 1-simplices in can be often interpreted as being homotopies, this category is often called the homotopy category of , and then the notation may be used.
This notions is closely related to the next, by using, say the hammock localisation? of Dwyer and Kan, as then of that simplicially enriched category, coincides with the following.
that sends every weak equivalence in to an isomorphism in .
One also writes or and calls it the localization of at the collection of weak equivalences.
More in detail, the universality of means the following:
The second condition implies that the functor in the first condition is unique up to unique isomorphism.
If it exists, the homotopy category is unique up to equivalence of categories.
As described at localization, in general, the morphisms of must be constructed using zigzags of morphisms in in which the backwards-pointing arrows are weak equivalences. This means that in general, need not be locally small even if is. However, in many cases (such as any model category) there is a more direct description of the morphisms in as homotopy classes of maps in between suitably “good” (fibrant and cofibrant) objects.
In 2-categorical terms, the homotopy category is the coinverter of the canonical 2-cell
where is the category whose objects are morphisms in and whose morphisms are commutative squares in .
Ho(Top) is often restricted to the full subcategory of spaces of the homotopy type of a CW-complex (the full subcategory of CW-complexes in ). This is equivalent to , the homotopy category of the standard Quillen-model structure on simplicial sets. This equivalence is one aspect of the homotopy hypothesis.
See the references at model category.