# Kerodon

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Example 1.3.1.4. 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 $d^{1}_0(f) = Y$ and the source $d^{1}_1(f) = X$, 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 $s^{0}_0(X)$ is the identity morphism $\operatorname{id}_{X}: X \rightarrow X$.