Paths and cylinders
The homotopy category of an (∞,1)-category is its decategorification to an ordinary category obtained by identifying 1-morphisms that are connected by a 2-morphism.
If the (∞,1)-category is presented by a category with weak equivalences (for instance as the simplicial localization ) then the notion of homotopy category of (where the weak equivalences are universally turned into isomorphisms) coinicides with that of :
The component-wise definition depend on the chosen model for -categories, as either
In terms of simplicially enriched categories
The homotopy category of a sSet-enriched category (equivalently of a Top-enriched category) is hom-wise the image under the functor
which sends each simplicial set to its 0th homotopy set of connected components, i.e. to the set of path components:
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 , which is the homotopy category of an -category of ∞Grpd.
Let 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 , which means that since every object in is cofibrant, it preserves weak equivalences in both arguments and hence descends to the homotopy category
This inuces a canonical functor which is given by the identity on objects and: . Then since , it is easy to see that .
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 -enriched categories.
Viewing as a simplicial set, the homotopy category can also be described as its fundamental category , i.e. the image of by the left adjoint of the nerve functor .
The Brown representability theorem characterizes representable functors on homotopy categories of -categories:
(Lurie “Higher Algebra”, theorem 126.96.36.199)