## You are here

Homenerve

## Primary tabs

# 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias