on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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 $(\infty,1)$-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 $C$ which is known in some way or other to be the $1$-categorical truncation of a higher category $\hat C$. Standard examples include the categories SimpSet of simplicial sets (or Top of topological spaces) and $Ch(Ab)$ 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 $(\infty,1)$-category $\hat C$ is canonically associated a 1-category $Ho(\hat C)$ called the homotopy category of an (infinity,1)-category, which is obtained from $\hat C$ not by simply forgetting the higher morphisms, but by quotienting them out, i.e. by remembering the equivalence classes of 1-morphisms. In the $(\infty,1)$-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 $(\infty,1)$-categories as homotopies. Therefore the name homotopy category of an $(\infty,1)$-category for $Ho(\hat C)$. In particular $Ho(\hat{Top})$ 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 $C$, but now equipped with the structure of a category with weak equivalences which indicates which morphisms in $C$ are to be regarded as equivalences in a higher categorical context, there is a universal solution to the problem of finding a category $Ho(C)$ equipped with a functor $Q : C \to Ho(C)$ such that $Q$ sends all (morphisms labeled as) weak equivalences in $C$ to isomorphisms in $Ho(C)$.
In good situations, one may also find an $(\infty,1)$-category $\hat C$ corresponding to $C$, and the notions of homotopy category $Ho(C)$ and $Ho(\hat C)$ then coincide.
This is, in particular, the case when $C$ is equipped with the structure of a combinatorial simplicial model category and $\hat C$ is the $(\infty,1)$-category presented by $C$ 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 $C$, we can form for each pair of objects, $x,y$, of objects of $C$, the set, $\pi_0C(x,y)$, of connected components of the ‘function space’ $C(x,y)$. As $\pi_0$ preserves finite limits, this gives a category, denoted $\pi_0(C)$. As 1-simplices in $C(x,y)$ can be often interpreted as being homotopies, this category $\pi_0(C)$ is often called the homotopy category of $C$, and then the notation $Ho(C)$ may be used.
This notions is closely related to the next, by using, say the hammock localisation? of Dwyer and Kan, as then $\pi_0$ of that simplicially enriched category, coincides with the following.
Given a category with weak equivalences (such as a model category), its homotopy category $Ho(C)$ is – if it exists – the category which is universal with the property that there is a functor
that sends every weak equivalence in $C$ to an isomorphism in $Ho(C)$.
One also writes $Ho(C) := W^{-1}C$ or $C[W^{-1}]$ and calls it the localization of $C$ at the collection $W$ of weak equivalences.
More in detail, the universality of $Ho(C)$ means the following:
The second condition implies that the functor $F_Q$ in the first condition is unique up to unique isomorphism.
If it exists, the homotopy category $Ho(C)$ is unique up to equivalence of categories.
As described at localization, in general, the morphisms of $Ho(C)$ must be constructed using zigzags of morphisms in $C$ in which the backwards-pointing arrows are weak equivalences. This means that in general, $Ho(C)$ need not be locally small even if $C$ is. However, in many cases (such as any model category) there is a more direct description of the morphisms in $Ho(C)$ as homotopy classes of maps in $C$ between suitably “good” (fibrant and cofibrant) objects.
In 2-categorical terms, the homotopy category $Ho(C)$ is the coinverter of the canonical 2-cell
where $W$ is the category whose objects are morphisms in $W$ and whose morphisms are commutative squares in $C$.
In classical homotopy theory, the homotopy category refers to the homotopy category Ho(Top) of Top with weak equivalences taken to be weak homotopy equivalences.
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 $Ho(Top)$). This is equivalent to $Ho(sSet_{Quillen})$, the homotopy category of the standard Quillen-model structure on simplicial sets. This equivalence is one aspect of the homotopy hypothesis.
In homological algebra the localization of the category of chain complexes at the quasi-isomorphisms is called the derived category. But see also at homotopy category of chain complexes.
In stable homotopy theory one considers the homotopy category of spectra.
See the references at model category.