Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 1.2.1.4. Let $\operatorname{\mathcal{C}}$ be a category. Then:

  • Vertices of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with objects of the category $\operatorname{\mathcal{C}}$.

  • Edges of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with morphisms in the category $\operatorname{\mathcal{C}}$.

  • Let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, regarded as an edge of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then the faces of $f$ are given by the target $d_0 f = Y$ and the source $d_1 f = X$, respectively.

  • Let $X$ be an object of $\operatorname{\mathcal{C}}$, which we regard as a vertex of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then the degenerate edge $s_0(X)$ is the identity morphism $\operatorname{id}_{X}: X \rightarrow X$.