Kerodon

Remark 1.3.1.12. The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$ from the category $\operatorname{Cat}$ of (small) categories to the category $\operatorname{Set_{\Delta }}$ of simplicial sets. This is a special case of the construction described in Variant 1.2.2.8. More precisely, we can identify $\operatorname{N}_{\bullet }$ with the functor $\operatorname{Sing}_{\bullet }^{Q}$, where $Q: \operatorname{{\bf \Delta }}\rightarrow \operatorname{Cat}$ is the functor which carries each object $[n] \in \operatorname{{\bf \Delta }}$ to itself, regarded as a category. It follows from Proposition 1.2.3.15 that this functor admits a left adjoint, which we will study in §1.3.6.