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

Idea

The homotopy category of an (infinity,1)-category $C$ is, roughly, the best 1-categorical approximation to $C$. It has the same objects as $C$, and its morphisms are equivalence classes of 1-morphisms in $C$.

Definition

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

Urs: something is missing here: the homotopy category of an $(\mathrm{\infty }\mathrm{,1})$-category is supposed to be enriched in ${\mathrm{Ho}}_{\mathrm{Top}}$, I think

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