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

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## 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 $𝒞$:

$\mathrm{Ho}\left(𝒞\right)\simeq \mathrm{Ho}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$Ho(\mathcal{C}) \simeq Ho(C) \,.

## Definition

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

## 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

${\pi }_{0}:\mathrm{sSet}\to \mathrm{Set}\phantom{\rule{thinmathspace}{0ex}},$\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:

${\mathrm{Hom}}_{hC}\left(A,B\right):={\pi }_{0}\left({\mathrm{Hom}}_{C}\left(A,B\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $\left(\infty ,1\right)$-category of ∞Grpd.

Let $h:\mathrm{sSet}\to \mathrm{Ho}\left(\mathrm{sSet}\right)$ 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

$\begin{array}{ccc}\mathrm{sSet}×\mathrm{sSet}& \stackrel{×}{\to }& \mathrm{sSet}\\ {↓}^{h×h}& & {↓}^{h}\\ \mathrm{Ho}\left(\mathrm{sSet}\right)×\mathrm{Ho}\left(\mathrm{sSet}\right)& \stackrel{h\left(×\right)}{\to }& \mathrm{Ho}\left(\mathrm{sSet}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $h:\mathrm{sSet}\mathrm{Cat}\to \mathrm{Ho}\left(\mathrm{sSet}\right)\mathrm{Cat}$ which is given by the identity on objects and: ${\mathrm{Map}}_{hC}\left(A,B\right):=h{\mathrm{Map}}_{C}\left(A,B\right)$. Then since ${\mathrm{Hom}}_{C}\left(A,B\right)={\mathrm{Hom}}_{\mathrm{sSet}}\left({\Delta }^{0},{\mathrm{Map}}_{C}\left(A,B\right)\right)$, it is easy to see that ${\mathrm{Hom}}_{\mathrm{hC}}\left(A,B\right)={\mathrm{Hom}}_{\mathrm{Ho}\left(\mathrm{sSet}\right)}\left(h{\Delta }^{0},h{\mathrm{Map}}_{C}\left(A,B\right)\right)={\pi }_{0}{\mathrm{Map}}_{C}\left(A,B\right)$.

## 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

$h:\mathrm{sSet}\to \mathrm{Cat}\phantom{\rule{thinmathspace}{0ex}},$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

Revised on October 6, 2011 12:59:17 by Rasmus Bentmann? (188.178.249.137)