nLab
homotopy category

Context

Homotopy theory

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

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 (,1)(\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 CC which is known in some way or other to be the 11-categorical truncation of a higher category C^\hat C. Standard examples include the categories SimpSet of simplicial sets (or Top of topological spaces) and Ch(Ab)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 (,1)(\infty,1)-category C^\hat C is canonically associated a 1-category Ho(C^)Ho(\hat C) called the homotopy category of an (infinity,1)-category, which is obtained from C^\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 (,1)(\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 (,1)(\infty,1)-categories as homotopies. Therefore the name homotopy category of an (,1)(\infty,1)-category for Ho(C^)Ho(\hat C). In particular Ho(Top^)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 CC, but now equipped with the structure of a category with weak equivalences which indicates which morphisms in CC 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)Ho(C) equipped with a functor Q:CHo(C)Q : C \to Ho(C) such that QQ sends all (morphisms labeled as) weak equivalences in CC to isomorphisms in Ho(C)Ho(C).

In good situations, one may also find an (,1)(\infty,1)-category C^\hat C corresponding to CC, and the notions of homotopy category Ho(C)Ho(C) and Ho(C^)Ho(\hat C) then coincide.

This is, in particular, the case when CC is equipped with the structure of a combinatorial simplicial model category and C^\hat C is the (,1)(\infty,1)-category presented by CC 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.

Definition (for a simplicially enriched category)

Given a simplicially enriched category CC, we can form for each pair of objects, x,yx,y, of objects of CC, the set, π 0C(x,y)\pi_0C(x,y), of connected components of the ‘function space’ C(x,y)C(x,y). As π 0\pi_0 preserves finite limits, this gives a category, denoted π 0(C)\pi_0(C). As 1-simplices in C(x,y)C(x,y) can be often interpreted as being homotopies, this category π 0(C)\pi_0(C) is often called the homotopy category of CC, and then the notation Ho(C)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 π 0\pi_0 of that simplicially enriched category, coincides with the following.

Definition (for a category with weak equivalences)

Given a category with weak equivalences (such as a model category), its homotopy category Ho(C)Ho(C) is – if it exists – the category which is universal with the property that there is a functor

Q:CHo(C) Q : C \to Ho(C)

that sends every weak equivalence in CC to an isomorphism in Ho(C)Ho(C).

One also writes Ho(C):=W 1CHo(C) := W^{-1}C or C[W 1]C[W^{-1}] and calls it the localization of CC at the collection WW of weak equivalences.

More in detail, the universality of Ho(C)Ho(C) means the following:

  • for any (possibly large) category AA and functor F:CAF : C \to A such that FF sends all wWw \in W to isomorphisms in AA, there exists a functor F Q:Ho(C)AF_Q : Ho(C) \to A and a natural isomorphism
C F A Q F Q Ho(C) \array{ C &&\stackrel{F}{\to}& A \\ \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q} \\ Ho(C) }

The second condition implies that the functor F QF_Q in the first condition is unique up to unique isomorphism.

Properties

  • If it exists, the homotopy category Ho(C)Ho(C) is unique up to equivalence of categories.

  • As described at localization, in general, the morphisms of Ho(C)Ho(C) must be constructed using zigzags of morphisms in CC in which the backwards-pointing arrows are weak equivalences. This means that in general, Ho(C)Ho(C) need not be locally small even if CC is. However, in many cases (such as any model category) there is a more direct description of the morphisms in Ho(C)Ho(C) as homotopy classes of maps in CC between suitably “good” (fibrant and cofibrant) objects.

  • In 2-categorical terms, the homotopy category Ho(C)Ho(C) is the coinverter of the canonical 2-cell

    W C \array{& \to \\ W & \Downarrow & C\\ & \to}

    where WW is the category whose objects are morphisms in WW and whose morphisms are commutative squares in CC.

Examples

References

See the references at model category.

Revised on August 24, 2013 15:32:57 by Tim Porter (95.147.236.162)