equivalences in/of $(\infty,1)$-categories
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 $\mathcal{C}$:
The component-wise definition depend on the chosen model for $(\infty,1)$-categories, as either
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
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 $sSet_{Quillen}$, 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
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 $Hom_{hC}(A,B)=Hom_{Ho(sSet)}(\mathbf{h} \Delta^0, \mathbf{h} Map_C(A,B))=\pi_0 Map_C(A,B)$.
Similar, but more complicated, definitions work for complete Segal spaces and Segal 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 $N$.
The Brown representability theorem characterizes representable functors on homotopy categories of $(\infty,1)$-categories:
Let $\mathcal{C}$ be a locally presentable (∞,1)-category, generated by a set
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
(from the opposite of the homotopy category of $\mathcal{C}$ to Set)
is representable precisely if it satisfies these two conditions:
$F$ sends small coproducts to products;
$F$ sends (∞,1)-pushouts $X \underset{Z}{\sqcup}Y$ to epimorphisms, i.e. the canonical morphisms into the fiber product
are surjections.
(Lurie “Higher Algebra”, theorem 1.4.1.2)
Jacob Lurie, section 1.4.1 of Higher Algebra