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

Idea

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

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

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

Definition

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

• simplicially enriched category/ topologically enriched categories

• complete Segal space

• Segal category

• quasi-category

In terms of simplicially enriched categories

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

$$\pi_0 : sSet \to Set \,,$$

which sends each simplicial set to its 0th homotopy set of connected components, i.e. to the set of path components:

$$Hom_{h C}(A,B) := \pi_0(Hom_C(A,B)) \,.$$

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 $sSet_{Quillen}$, which is the homotopy category of an $(\infty,1)$-category of ∞Grpd.

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

$$\array{ sSet \times sSet &\stackrel{\times}{\to}& sSet \\ \downarrow^{\mathrlap{\mathbf{h} \times \mathbf{h}}} && \downarrow^{\mathrlap{\mathbf{h}}} \\ Ho(sSet) \times Ho(sSet) &\stackrel{\mathbf{h}(\times)}{\to}& Ho(sSet) } \,.$$

This inuces a canonical functor $h:sSet Cat\to Ho(sSet) Cat$ which is given by the identity on objects and: $Map_{h C}(A,B):=\mathbf{h} Map_{C}(A,B)$. Then since $Hom_C(A,B)=Hom_{sSet}(\Delta^0,Map_{C}(A,B))$, it is easy to see that $Hom_{hC}(A,B)=Hom_{Ho(sSet)}(\mathbf{h} \Delta^0, \mathbf{h} Map_C(A,B))=\pi_0 Map_C(A,B)$.

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 $sSet$-enriched categories.

Viewing $C$ as a simplicial set, the homotopy category $hC$ can also be described as its fundamental category $\tau_1(C)$, i.e. the image of $C$ by the left adjoint $\tau_1 : SSet \to Cat$ of the nerve functor $N$.

Properties

Brown representability

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

Proposition

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

$$\{S_i \in \mathcal{C}\}_{i \in I}$$

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

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

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

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

is representable precisely if it satisfies these two conditions:

1. $F$ sends small coproducts to products;

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

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

   are surjections.

(Lurie “Higher Algebra”, theorem 1.4.1.2)

References

• Jacob Lurie, Section 1.2.3, p. 33 of Higher Topos Theory

• Jacob Lurie, section 1.4.1 of Higher Algebra