---
tag: JTAKAPTT8H
kind: concept
name: Stalk
msc: [54B40, 14F06]
sources:
  - url: https://en.wikipedia.org/wiki/Stalk_(sheaf)
same_as:
  wikidata: Q17103642
  wikipedia: Stalk (sheaf)
---

## Summary

The stalk of a sheaf at a point is the set of germs of sections at that point.
A germ is a section defined near the point, where two sections are identified when they agree on some smaller neighbourhood.
Stalks capture the local behaviour of a sheaf at a single point.
Many properties of sheaves and of maps between them, such as being zero, injective or surjective, can be checked stalk by stalk.

## Explanations

### L1 human

Zoom in on one point of a space and look only at what a sheaf does very close to it.
Two sections that agree on some small enough neighbourhood are counted as the same.
For the sheaf of continuous functions, the stalk at a point is the set of germs of continuous functions there.
For holomorphic functions near $0$ in $\C$, a germ is the same as a power series with a positive radius of convergence.