homotopy category of an (infinity,1)-category


(,1)(\infty,1)-Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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) \,.


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

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:sSetSet, \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:sSetHo(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:sSetCatHo(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:SSetCat\tau_1 : SSet \to Cat of the nerve functor NN.


Brown representability

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


Let 𝒞\mathcal{C} be a locally presentable (∞,1)-category, generated by a set

{S i𝒞} iI \{S_i \in \mathcal{C}\}_{i \in I}

of compact objects (i.e. every object of 𝒞\mathcal{C} is an (∞,1)-colimit of the objects S iS_i.)

If each S iS_i admits the structure of a cogroup object in the homotopy category Ho(𝒞)Ho(\mathcal{C}), then a functor

F:Ho(𝒞) opSet F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set

(from the opposite of the homotopy category of 𝒞\mathcal{C} to Set)

is representable precisely if it satisfies these two conditions:

  1. FF sends small coproducts to products;

  2. FF sends (∞,1)-pushouts XZYX \underset{Z}{\sqcup}Y to epimorphisms, i.e. the canonical morphisms into the fiber product

    F(XZY)epiF(X)×F(Z)F(Y) F\left(X \underset{Z}{\sqcup}Y\right) \stackrel{epi}{\longrightarrow} F(X) \underset{F(Z)}{\times} F(Y)

    are surjections.

(Lurie “Higher Algebra”, theorem


Revised on March 11, 2016 07:59:50 by Urs Schreiber (