homotopy category


Homotopy theory

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Given a category with weak equivalences (𝒞,W)(\mathcal{C},W), then its homotopy category Ho(𝒞)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

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

The classical example is the category of topological spaces with weak equivalences 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 homoemorphic to CW-complexes with left homotopy-classes of continuous functions between them, whence the name “homotopy category”.

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 at homotopy category of a model category for more on this.

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

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


For simplicially enriched categories

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.

For categories 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.


  • 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.



See the references at model category.

Revised on March 17, 2017 17:20:18 by Urs Schreiber (