nLab
homotopy category

Idea

Frequently 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 $\mathrm{Ch}(\mathrm{Ab})$ 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 $(\mathrm{\infty },1)$-category $\hat{C}$ is canonically associated a 1-category $\mathrm{Ho}(\hat{C})$ 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 $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-categories as homotopies. Therefore the name homotopy category of an $(\mathrm{\infty },1)$-category for $\mathrm{Ho}(\hat{C})$. In particular $\mathrm{Ho}(\widehat{\mathrm{Top}})$ is the standard homotopy category originally introduced in topology.

Now, given just the truncated 1-category $C$ but 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 $\mathrm{Ho}(C)$ equipped with a functor $Q:C\to \mathrm{Ho}(C)$ such that $Q$ sends all (morphisms labeled as) weak equivalences in $C$ to isomorphisms in $\mathrm{Ho}(C)$.

In good situations, one may also find an $(\mathrm{\infty },1)$-category $\hat{C}$ corresponding to $C$, and the notions of homotopy category $\mathrm{Ho}(C)$ and $\mathrm{Ho}(\hat{C})$ 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 $(\mathrm{\infty },1)$-category presented by $C$ with its model structure. (For instance HTT, remark A.3.1.8).

Definition

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

$$Q:C\to \mathrm{Ho}(C)$$Q : C \to Ho(C)

that sends every weak equivalence in $C$ to an isomorphism in $\mathrm{Ho}(C)$.

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

More in detail, the universality of $\mathrm{Ho}(C)$ means the following:

• for any (possibly large) category $A$ and functor $F:C\to A$ such that $F$ sends all $w\in W$ to isomorphisms in $A$, there exists a functor $F_Q:\mathrm{Ho}(C)\to A$ and a natural isomorphism

$$\begin{array}{cccc}C& & \stackrel{F}{\to }& A\\ {\downarrow }^Q& {\Downarrow }^{\simeq }& {\nearrow }_{F_Q}\\ \mathrm{Ho}(C)\end{array}$$\array{ C &&\stackrel{F}{\to}& A \\ \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q} \\ Ho(C) }

• the functor $Q^{*}:\mathrm{Func}(\mathrm{Ho}(C),A)\to \mathrm{Func}(C,A)$ is a full and faithful functor.

The second condition implies that the functor $F_Q$ in the first condition is unique up to unique isomorphism.

Harry: I think there should be a little more discussion of the classical case (i.e. pointed topological spaces). There’s a functor taking this category to its homotopy category in the following way: It sends $\mathrm{Hom}((X,x_0),(Y,y_0))$ to $[(X,x_0);(Y,y_0)]$. However, the is a little more to it, so to speak, because I think we can describe $[(X,x_0);(Y,y_0)]$ as a coequalizer of $\mathrm{Hom}(-\times I,-)$=>$\mathrm{Hom}(-,-)$, which gives us the appropriate idea fof “modulo homotopy”. This is at least how it works if the second coordinate is fixed. However, since it is a bifunctor, I’m not sure if I need to be more careful/not of taking colimits. To be precise, this takes place in the category of bipointed topological spaces (because we want to attach X at two points to the interval (this is how we get the two arrows). I’d appreciate it if someone would clarify this for me (so I could write up the section classical definition).

Tim Do you mean $\mathrm{Hom}(-\times I,-)$=>$\mathrm{Hom}(-,-)$, which make sense and is one otf the usual ways of defining this. (You can find this in Kamps and Porter’s book for example (I would say that wouldn’t I! but the royalties are really measily, and when there has been conversion of currencies followed by tax, etc… !)) I don’t think you need bipointed spaces, but you do need to be more careful about the product, if you are working in pointed spaces.

Properties

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

• As described at localization, in general, the morphisms of $\mathrm{Ho}(C)$ must be constructed using zigzags of morphisms in $C$ in which the backwards-pointing arrows are weak equivalences. This means that in general, $\mathrm{Ho}(C)$ 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 $\mathrm{Ho}(C)$ as homotopy classes of maps in $C$ between suitably “good” (fibrant and cofibrant) objects.

• In 2-categorical terms, the homotopy category $\mathrm{Ho}(C)$ is the coinverter of the canonical 2-cell

  $$\begin{array}{cc}& \to \\ W& \Downarrow & C\\ & \to \end{array}$$\array{& \to \\ W & \Downarrow & C\\ & \to}

  where $W$ is the category whose objects are morphisms in $W$ and whose morphisms are commutative squares in $C$.

Examples

• 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 $\mathrm{Ho}(\mathrm{Top})$). This is equivalent to $\mathrm{Ho}({\mathrm{sSet}}_{\mathrm{Quillen}})$, 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.

• For the homotopy category of Cat, see Ho(Cat).

Related concepts

• homotopy category of an (infinity,1)-category

• homotopy 2-category

References

See the references at model category.