Kerodon

Example 4.6.1.4. Let $\operatorname{\mathcal{C}}$ be an ordinary category containing objects $X$ and $Y$, which we will identify with objects of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then the morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( X, Y)$ of Construction 4.6.1.1 can be identified with the constant simplicial set having the value $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Example 4.6.4.6). In particular, when $X = Y$ we can identify the simplicial set $\operatorname{End}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }(X) = \operatorname{Hom}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( X, X)$ with the endomorphism monoid $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ of Example 1.3.2.2.