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# nerve

Let $\mathord{\mathbf{Set}}$ be the category of all sets with functions as the morphisms, and let $\mathord{\mathbf{Cat}}$ 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 $i\colon\Delta\to\mathord{\mathbf{Cat}}$ is the fully faithful functor that takes each ordered set $[n]$ in the simplicial category, $\Delta$, to the pre-order $\mathord{\mathbf{n+1}}$. The nerve is a functor $\mathord{\mathbf{Cat}}\to\mathord{\mathbf{Set}}^{{\Delta^{\mathrm{op}}}}$.

###### Example 1 (Nerve of an open covering)

Let $X$ be a topological space with open cover $\{U_{\alpha}\}$. The nerve of the open covering of $X$ is the nerve of the partially-ordered set $\{U_{\alpha}\}$ with relation that of inclusion. Thus, it assigns to every $n$ the set of maps from the totally ordered set $n+1$ to the poset $\{U_{\alpha}\}$.

## Mathematics Subject Classification

18G30*no label found*

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