nLab homotopy category

Idea

Given a category with weak equivalences $(\mathcal{C},W)$, its homotopy category $Ho(\mathcal{C})$ is, if it exists, the result of universally forcing the weak equivalences to become actual isomorphisms, also called the localization at the weak equivalences

$$\mathcal{C} \longrightarrow Ho(\mathcal{C})= \mathcal{C}[W^{-1}] \,.$$

The classical example is the category of topological spaces with weak equivalences being those continuous functions which are homotopy equivalences or weak homotopy equivalences. The corresponding homotopy category is often referred to as “the homotopy category” by default, or the “classical homotopy category” for emphasis. This turns out to be equivalent to the category of topological spaces or (for weak homotopy equivalences) of just those homeomorphic to CW-complexes with left homotopy-classes of continuous functions between them, whence the name “homotopy category” comes.

The existence of a homotopy category as well as tractable presentations of it typically require extra properties of the class of weak equivalences (such as that they admit a calculus of fractions) or even extra structure (such as fibration category/cofibration category structure, or full model category structure, or further enhancements of that to simplicial model category structures, etc). See homotopy category of a model category for more on this.

More generally, associated to every (∞,1)-category is a homotopy category whose morphisms are literally the homotopy classes of the original morphisms. See homotopy category of an (∞,1)-category for more on this.

These two concepts of a homotopy category are compatible: associated to a category with weak equivalences is, if it exists, an (∞,1)-category obtained by universally forcing the weak equivalences to become actual homotopy equivalences, also called the simplicial localization $L_W \mathcal{C}$ at the weak equivalences. The homotopy categories of $(\mathcal{C},W)$ and of $L_W \mathcal{C}$ coincide, which justifies the terminology “homotopy category” generally.

Definition

For simplicially enriched categories

Given a simplicially enriched category $C$, we can form, for each pair of objects $x$ and $y$ 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 notion 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.

For categories with weak equivalences

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

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

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.

In more detail, the universality of $Ho(C)$ 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 $F_Q : Ho(C) \to A$ and a natural isomorphism

$$\array{ C &&\stackrel{F}{\to}& A \\ \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q} \\ Ho(C) }$$

• the functor $Q^* : Func(Ho(C),A) \to Func(C,A)$ 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.

Properties

• 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 in 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

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

  where $W$ is the category whose objects are morphisms in $C$ 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 $Ho(Top)$). 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 homotopy category of chain complexes.

• In stable homotopy theory, one considers the stable homotopy category of spectra.

• In equivariant stable homotopy theory, one considers the equivariant stable homotopy category of spectra.

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

• For the homotopy category of that of combinatorial model categories, see Ho(CombModCat).

Related concepts

• homotopy category of a model category

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

• homotopy 2-category

• simplicial homotopy category

References

Early discussion:

• Edgar Brown, Abstract homotopy theory, Trans. AMS 119 no. 1 (1965) $[$doi:10.1090/S0002-9947-1965-0182970-6$]$

For more, see the references at model category or classical homotopy category.