Definition 1.4.1.1. Let $\operatorname{\mathcal{C}}= S_{\bullet }$ be an $\infty $-category. An object of $\operatorname{\mathcal{C}}$ is a vertex of the simplicial set $S_{\bullet }$ (that is, an element of the set $S_{0}$). A morphism of $\operatorname{\mathcal{C}}$ is an edge of the simplicial set $S_{\bullet }$ (that is, an element of $S_1$). If $f \in S_{1}$ is a morphism of $\operatorname{\mathcal{C}}$, we will refer to the object $X = d^{1}_1(f)$ as the source of $f$ and to the object $Y = d^{1}_0(f)$ as the target of $f$. In this case, we will say that $f$ is a morphism from $X$ to $Y$. For any object $X$ of $\operatorname{\mathcal{C}}$, we can regard the degenerate edge $s^{0}_0(X)$ as a morphism from $X$ to itself; we will denote this morphism by $\operatorname{id}_{X}$ and refer to it as the identity morphism of $X$.
1.4.1 Objects and Morphisms
We begin by introducing some terminology.
Notation 1.4.1.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often write $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an object of $\operatorname{\mathcal{C}}$. We use the phrase “$f: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$” to indicate that $f$ is a morphism of $\operatorname{\mathcal{C}}$ having source $X$ and target $Y$.
Example 1.4.1.3. Let $\operatorname{\mathcal{C}}$ be an ordinary category, and regard the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ as an $\infty $-category. Then:
The objects of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.
The morphisms of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ are the morphisms of $\operatorname{\mathcal{C}}$. Moreover, the source and target of a morphism of $\operatorname{\mathcal{C}}$ coincide with the source and target of the corresponding morphism in $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
For every object $X \in \operatorname{\mathcal{C}}$, the identity morphism $\operatorname{id}_{X}$ does not depend on whether we view $X$ as an object of the category $\operatorname{\mathcal{C}}$ or the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
Example 1.4.1.4. Let $X$ be a topological space, and regard the simplicial set $\operatorname{Sing}_{\bullet }(X)$ as an $\infty $-category. Then:
The objects of $\operatorname{Sing}_{\bullet }(X)$ are the points of $X$.
The morphisms of $\operatorname{Sing}_{\bullet }(X)$ are continuous paths $f: [0,1] \rightarrow X$. The source of a morphism $f$ is the point $f(0)$, and the target is the point $f(1)$.
For every point $x \in X$, the identity morphism $\operatorname{id}_{x}$ is the constant path $[0,1] \rightarrow X$ taking the value $x$.
Definition 1.4.1.5 (Endomorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An endomorphism in $\operatorname{\mathcal{C}}$ is a morphism $f: X \rightarrow X$ of $\operatorname{\mathcal{C}}$ for which the source and target of $f$ are the same. In this case, we will say that $f$ is an endomorphism of $X$.