Between any two objects $x,y$ in an (∞,1)-category $C$ there is an ∞-groupoid of morphisms. It is well-defined up to equivalence. When the $(\infty,1)$-category is incarnated as a quasi-category, there are several explicit ways to extract representatives of this hom-object.
Let $C$ be a quasi-category (or any simplicial set), and $x,y \in C_0$ any two objects. Then write
where
$\tau$ is the left adjoint to the homotopy coherent nerve;
$\tau(C)$ is accordingly the simplicially enriched category incarnation of $C$,
$\tau(C)(x,y)$ is the sSet-hom-object of that $sSet$-enriched category;
$[\tau(C)(x,y)]$ is the equivalence class of this object.
This defines $Hom_C(x,y)$ as an equivalence class of $\infty$-groupoids, but at the same time defines a particular representative: if $C$ is a quasi-category then $\tau(C)(x,y)$ is a Kan complex that represents this class.
This is useful for many purposes, but $\tau(C)$ is usually hard to compute explicitly. The following three other definitions of representatives of $Hom_C(x,y)$ are often useful.
Definition
For $C$ and $x,y$ as above, write
for the pullback
in sSet of the path object $C^{\Delta[1]}$ (the cartesian internal hom in sSet with the 1-simplex $\Delta[1]$) .
Write
for the simplicial set whose $n$-simplices are defined to be those morphisms $\sigma : \Delta[n+1] \to C$ such that the restriction to $\Delta\{0, \cdots, n\}$ is the constant map to $x$ and the restriction to $\Delta\{n+1\}$ is the map to $y$.
Analogously, write
for the simplicial set whose $n$-simplices are morphisms $\Delta[n+1] \to X$ which restrict to $x$ on $\{0\}$ and are constant on $y$ when restricted to $\{1, \cdots, n+1\}$.
Remark The 1-cells in $Hom_C^R(x,y)$, $Hom_C^L(x,y)$ and $Hom_C^{L R}(x,y)$ are 2-globes in $C$. The 2-cells are commuting squares of vertical composites of 2-globes forming a 3-globe.
Equivalently this may be understood in terms of fibers of over quasi-categories.
Recall that for $p : K \to C$ a morphism, we have the over quasi-category $C_{/p}$ defined by
where on the right we have the set of morphisms in $sSet$ out of the join of simplicial sets that restrict on $K$ to $p$.
This comes with the canonical projection $C^{p/} \to C$, which sends $(\Delta[n] \star K \to C)$ to the restriction $(\Delta[n] \to \Delta[n] \star K \to C)$.
There is also the other, equivalent, definition $C^{/p}$ of over quasi-category, defined using the other, equivalent, definition $\diamondsuit$ of join of quasi-categories by
Obervation
We have
where on the right we have the pullback in sSet in the diagram
and the equality sign denotes an isomorphism in sSet.
And we have
Proof For the first statement use the identification of $\Delta[n+1]$ with the join of simplicial sets $\Delta[n] \star \Delta[0]$, as described there.
For the second statement use that $\Delta[n] \diamondsuit \Delta[0]$ is the colimit $\Delta[n+1]$ in
so that
because
Remark One advantage of the representative $\tau(C)(c,y)$ of $Hom_C(x,y)$ is that, by the fact that $\tau(C)$ is an sSet-enriched category, there is a strict composition operation
This is not available for the $Hom_C^R(x,y)$ and $Hom_C^L(x,y)$
Proposition
If the simplicial set $C$ is a quasi-category, then $Hom_C^R(x,y)$ is a Kan complex.
Proof
This is HTT, prop 1.2.2.3.
From the definition it is clear that $Hom_C^R(x,y)$ has fillers for all inner and right outer horns $\Lambda[n]_{1 \leq i \leq n}$, because these yield inner horns in $\Delta[n+1] = \Delta[n] \star \Delta[0]$. The claim follows then with the fact that every right fibration over the point is a Kan complex, as described there.
Proposition
If $C$ is a quasi-category then the canonical inclusions
are homotopy equivalences of Kan complexes.
Proof
This is HTT, cor. 4.2.1.7.
As described at join of quasi-categories the canonical morphism $C_{/y} \to C^{/y}$ is an equivalence of quasi-categories. So for the statement for $Hom_C^R(x,y)$ it suffices to show that this induces an equivalence of fibers over $C$. This follows from the fact that both $C_{/y} \to C$ and $C^{/y} \to C$ are Cartesian fibrations.
See Cartesian fibrations for these statements. This is HTT, prop. 3.3.1.5. (2).
The statement for $Hom_C^L(x,y)$ follows dually.