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

Idea

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

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

$$\mathrm{Ho}(𝒞)\simeq \mathrm{Ho}(C).$$Ho(\mathcal{C}) \simeq Ho(C) \,.

Definition

The details of the definition depend on the chosen model for $(\mathrm{\infty },1)$-categories, as either

• simplicially enriched category/ Top-category

• complete Segal space

• Segal category

• quasi-category

For simplicially enriched categories

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

$${\pi }_0:\mathrm{sSet}\to \mathrm{Set},$$\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 component?s:

$${\mathrm{Hom}}_{hC}(A,B):={\pi }_0({\mathrm{Hom}}_C(A,B)).$$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 ${\mathrm{sSet}}_{\mathrm{Quillen}}$, which is the homotopy category of an $(\mathrm{\infty },1)$-category of ∞Grpd.

Let $h:\mathrm{sSet}\to \mathrm{Ho}(\mathrm{sSet})$ 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 $\mathrm{sSet}$, which means that since every object in $\mathrm{sSet}$ is cofibrant, it preserves weak equivalences in both arguments and hence descends to the homotopy category

$$\begin{array}{ccc}\mathrm{sSet}\times \mathrm{sSet}& \stackrel{\times }{\to }& \mathrm{sSet}\\ {\downarrow }^{h\times h}& & {\downarrow }^h\\ \mathrm{Ho}(\mathrm{sSet})\times \mathrm{Ho}(\mathrm{sSet})& \stackrel{h(\times )}{\to }& \mathrm{Ho}(\mathrm{sSet})\end{array}.$$\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 $h:\mathrm{sSet}\mathrm{Cat}\to \mathrm{Ho}(\mathrm{sSet})\mathrm{Cat}$ which is given by the identity on objects and: ${\mathrm{Map}}_{hC}(A,B):=h{\mathrm{Map}}_C(A,B)$. Then since ${\mathrm{Hom}}_C(A,B)={\mathrm{Hom}}_{\mathrm{sSet}}({\Delta }^0,{\mathrm{Map}}_C(A,B))$, it is easy to see that ${\mathrm{Hom}}_{\mathrm{hC}}(A,B)={\mathrm{Hom}}_{\mathrm{Ho}(\mathrm{sSet})}(h{\Delta }^0,h{\mathrm{Map}}_C(A,B))={\pi }_0{\mathrm{Map}}_C(A,B)$.

For complete Segal spaces and Segal categories

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

For quasi-categories

For quasi-categories, one can write down a definition similar to those of $\mathrm{sSet}$-enriched categories, but there is also the following direct construction:

the simplicial nerve functor $N:$ Cat $\to$ sSet has a left adjoint

$$h:\mathrm{sSet}\to \mathrm{Cat},$$h : sSet \to Cat \,,

and the homotopy category of a quasi-category $C$ (a simplicial set with extra properties) is its image $hC$ under this functor.

References

Section 1.2.3, p. 33 of

• Jacob Lurie, Higher Topos Theory