homotopy category of an (infinity,1)-category (changes) in nLab

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Idea

The homotopy category of an (∞,1)-category 𝒞\mathcal{C} is its decategorification to an ordinary category obtained by identifying 1-morphisms that are connected by a 2-morphism.

If the (∞,1)-category 𝒞\mathcal{C} is presented by a category with weak equivalences CC (for instance as the simplicial localization 𝒞=LC\mathcal{C} = L C) then the notion of homotopy category of CC (where the weak equivalences are universally turned into isomorphisms) coinicides with that of 𝒞\mathcal{C}:

Ho(𝒞)≃Ho(C). Ho(\mathcal{C}) \simeq Ho(C) \,.

Definition

The component-wise definition depend on the chosen model for (∞,1)(\infty,1)-categories, as either

• simplicially enriched category / Top-categorytopologically enriched categories

• complete Segal space

• Segal category

• quasi-category

In terms of simplicially enriched categories

The homotopy category hCh C of a sSet-enriched category CC (equivalently of a Top-enriched category) is hom-wise the image under the functor

π 0:sSet→Set, \pi_0 : sSet \to Set \,,

which sends each simplicial set to its 0th homotopy set of connected components, i.e. to the set of path components:

Hom hC(A,B):=π 0(Hom C(A,B)). Hom_{h C}(A,B) := \pi_0(Hom_C(A,B)) \,.

Sometimes it is useful to regard this after all as an enriched category, but now enriched over the homotopy category of the standard model structure on simplicial sets sSet QuillensSet_{Quillen}, which is the homotopy category of an (∞,1)(\infty,1)-category of ∞Grpd.

Let h:sSet→Ho(sSet)\mathbf{h}: sSet \to Ho(sSet) be the localization functor to the homotopy category (of a category with weak equivalences). This functor is a monoidal functor by the fact that the cartesian monoidal product is a left-Quillen bifunctor for sSetsSet, which means that since every object in sSetsSet is cofibrant, it preserves weak equivalences in both arguments and hence descends to the homotopy category

sSet×sSet →× sSet ↓ h×h ↓ h Ho(sSet)×Ho(sSet) →h(×) Ho(sSet). \array{ sSet \times sSet &\stackrel{\times}{\to}& sSet \\ \downarrow^{\mathrlap{\mathbf{h} \times \mathbf{h}}} && \downarrow^{\mathrlap{\mathbf{h}}} \\ Ho(sSet) \times Ho(sSet) &\stackrel{\mathbf{h}(\times)}{\to}& Ho(sSet) } \,.

This inuces a canonical functor h:sSetCat→Ho(sSet)Cath:sSet Cat\to Ho(sSet) Cat which is given by the identity on objects and: Map hC(A,B):=hMap C(A,B)Map_{h C}(A,B):=\mathbf{h} Map_{C}(A,B). Then since Hom C(A,B)=Hom sSet(Δ 0,Map C(A,B))Hom_C(A,B)=Hom_{sSet}(\Delta^0,Map_{C}(A,B)), it is easy to see that Hom hC(A,B)=Hom Ho(sSet)(hΔ 0,hMap C(A,B))=π 0Map C(A,B)Hom_{hC}(A,B)=Hom_{Ho(sSet)}(\mathbf{h} \Delta^0, \mathbf{h} Map_C(A,B))=\pi_0 Map_C(A,B).

In terms of complete Segal spaces and Segal categories

Similar, but more complicated, definitions work for complete Segal spaces and Segal categories.

In terms of quasi-categories

For quasi-categories, one can write down a definition similar to those of sSetsSet-enriched categories.

Viewing CC as a simplicial set, the homotopy category hChC can also be described as its fundamental category τ 1(C)\tau_1(C), i.e. the image of CC by the left adjoint τ 1:SSet→Cat\tau_1 : SSet \to Cat of the nerve functor NN.

Properties

Brown representability

The Brown representability theorem characterizes representable functors on homotopy categories of (∞,1)(\infty,1)-categories:

(Lurie “Higher Algebra”, theorem 1.4.1.2)

References

• Jacob Lurie, Section 1.2.3, p. 33 of Higher Topos Theory

• Jacob Lurie, section 1.4.1 of Higher Algebra