Let $\mathcat{Set}$ be the category of all sets with functions as the morphisms, and let $\mathcat{Cat}$ be the category of all small categories with functors as the morphisms.

The nerve of a (small) category $C$ is the simplicial set $\hom(i(-),C)$ , where $i\colon \Delta \to \mathcat{Cat}$ is the fully faithful functor that takes each ordered set $[n]$ in the simplicial category, $\Delta$ , to the pre-order $\mathcat{n+1}$ . The nerve is a functor $\mathcat{Cat} \to \mathcat{Set}^{\Delta^\op}$ .