---
tag: 4P3S0DRZXB
kind: concept
name: Nerve of a cover
msc: [55U10, 55N05]
sources:
  - url: https://en.wikipedia.org/wiki/Nerve_complex
same_as:
  wikidata: Q1424012
  wikipedia: Nerve complex
---

## Summary

The nerve of a cover of a space is the [[simplicial complex|V8GZ0W1ZZ5]] that records which members of the cover meet.
It has one vertex for each nonempty member, and a set of members spans a simplex when they have a common point.
By the nerve theorem, a good cover has a nerve with the same homotopy type as the space.
This turns topology into combinatorics, and nerves are also the basis of Čech cohomology.

## Explanations

### L1 human

Cover a space with overlapping patches.
Draw a dot for each patch, an edge between two dots when the patches overlap, a triangle when three patches share a point, and so on.
The resulting shape is the nerve.
Cover a circle by three arcs, each overlapping the next, with no point in all three.
The nerve is the boundary of a triangle, which is again a circle.