PlanetMath: nerve

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nerve

(Definition)

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

"nerve" is owned by mhale.

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