The $0$-simplices of $\operatorname{Map}_{\mathcal{C}}(x, y)$ are precisely the morphisms from $x$ to $y$ in $\mathcal{C}$: the $1$-simplices $f \in \mathcal{C}_1$ with $d_1(f) = x$ and $d_0(f) = y$.

$\pi_0(\operatorname{Map}_{\mathcal{C}}(x, y)) = \operatorname{Hom}_{\operatorname{h}\mathcal{C}}(x, y)$

where $\operatorname{h}\mathcal{C}$ is the homotopy category of $\mathcal{C}$.

A $1$-simplex in $\operatorname{Map}_{\mathcal{C}}(x, y)$ is a homotopy between two morphisms $f, g: x \to y$. Concretely, it is a $2$-simplex $\sigma \in \mathcal{C}_2$ with $d_2(\sigma) = \mathrm{id}_x$ (or $\mathrm{id}_y$, depending on the model), $d_0(\sigma) = g$ and $d_1(\sigma) = f$.

Two morphisms are in the same connected component of $\operatorname{Map}_{\mathcal{C}}(x, y)$ if and only if they are homotopic.

A $2$-simplex in $\operatorname{Map}_{\mathcal{C}}(x, y)$ is a homotopy between homotopies. This higher structure is essential: while the homotopy category only sees $\pi_0$ of the mapping spaces, the full mapping space retains information about $\pi_1, \pi_2, \ldots$

For instance, in $\mathcal{S}$ (the $\infty$-category of spaces), $\operatorname{Map}_{\mathcal{S}}(X, Y) \simeq \operatorname{Map}(X, Y)$, the topological mapping space. Its higher homotopy groups $\pi_n(\operatorname{Map}(X, Y), f)$ record obstructions to deforming maps.

For a nerve $N(\mathcal{C})$, the mapping space $\operatorname{Map}_{N(\mathcal{C})}(x, y)$ is the discrete simplicial set $\operatorname{Hom}_{\mathcal{C}}(x, y)$: a set with no higher homotopical information. This is because $N(\mathcal{C})$ has unique inner horn fillers, so there is no room for non-trivial homotopies between morphisms.

$\pi_0(\operatorname{Map}_{N(\mathcal{C})}(x, y)) = \operatorname{Hom}_{\mathcal{C}}(x, y)$ and $\pi_n = 0$ for $n \geq 1$.

For a Kan complex $X$, the mapping space $\operatorname{Map}_X(x, y)$ is the space of paths from $x$ to $y$. This is the based loop space $\Omega_{x,y} X$ when $x = y$: $\operatorname{Map}_X(x, x) \simeq \Omega_x X$.

$\pi_n(\operatorname{Map}_X(x, x)) \cong \pi_{n+1}(X, x)$

So the mapping space in a Kan complex records the shifted homotopy groups.

In the $\infty$-category $\mathcal{S}$ of spaces, the mapping space is the usual mapping space:

$\operatorname{Map}_{\mathcal{S}}(X, Y) \simeq \operatorname{Map}(X, Y)$

For $X = S^1$ and $Y = S^2$: $\operatorname{Map}(S^1, S^2) \simeq \Omega S^2 \times S^2$, which has $\pi_0 = \mathbb{Z}$ (winding number is trivial for maps $S^1 \to S^2$, so only one component) and higher homotopy groups related to those of $S^2$.

For $X = S^n$ and $Y = K(\mathbb{Z}, n)$: $\operatorname{Map}(S^n, K(\mathbb{Z}, n)) \simeq \mathbb{Z} \times K(\mathbb{Z}, n)$, with $\pi_0 = \mathbb{Z}$ classifying cohomology classes.

For the $\infty$-category $D(R)$ (derived $\infty$-category of $R$-modules), the mapping space between complexes $C$ and $D$ is:

$\operatorname{Map}_{D(R)}(C, D) \simeq \operatorname{DK}(\tau_{\geq 0} \mathbf{R}\operatorname{Hom}_R(C, D))$

where $\operatorname{DK}$ is the Dold--Kan functor. The connected components give:

$\pi_0(\operatorname{Map}_{D(R)}(C, D)) = \operatorname{Hom}_{D(R)}(C, D) = H^0(\mathbf{R}\operatorname{Hom}(C, D))$ $\pi_n(\operatorname{Map}_{D(R)}(C, D)) = \operatorname{Ext}^{-n}_R(C, D)$

So the higher homotopy groups of the mapping space are the negative Ext groups.

For objects $x, y, z$ in a quasi-category $\mathcal{C}$, there is a composition map:

$\operatorname{Map}_{\mathcal{C}}(x, y) \times \operatorname{Map}_{\mathcal{C}}(y, z) \to \operatorname{Map}_{\mathcal{C}}(x, z)$

This map is well-defined up to coherent homotopy (not a strict map, but a map in the $\infty$-categorical sense). It is associative and unital up to coherent homotopy, making $\mathcal{C}$ into a category "enriched in spaces" in the homotopy-coherent sense.

This composition law is extracted from the $2$-simplex fillers in $\mathcal{C}$.

The homotopy category $\operatorname{h}\mathcal{C}$ of a quasi-category $\mathcal{C}$ is the ordinary category with:

$\operatorname{Ob}(\operatorname{h}\mathcal{C}) = \mathcal{C}_0, \quad \operatorname{Hom}_{\operatorname{h}\mathcal{C}}(x, y) = \pi_0(\operatorname{Map}_{\mathcal{C}}(x, y))$

Composition in $\operatorname{h}\mathcal{C}$ is induced by the composition of mapping spaces at the $\pi_0$ level. This is well-defined because the space of composites is connected (contractible, in fact).

The homotopy category loses information: it forgets the higher homotopy groups of the mapping spaces.