---
tag: KB76923A1M
kind: concept
name: Sheaf cohomology in topology
msc: [55N30, 14F06]
sources:
  - url: https://en.wikipedia.org/wiki/Sheaf_cohomology
---

## Summary

Sheaf cohomology assigns groups $H^q(X, F)$ to a topological space $X$ and a sheaf $F$ of abelian groups on $X$.
They are the right derived functors of the global sections functor.
It measures the failure of local sections to glue into global sections.
With a constant sheaf it recovers ordinary cohomology on reasonable spaces, and with other sheaves it gives local coefficients, Borel–Moore theory and Poincaré–Verdier duality.
The general notion is sheaf cohomology in algebraic geometry and topology.

## Explanations

### L1 human

A sheaf is a rule that tells you what the allowed local data are on each open piece of a space, such as continuous functions or locally constant functions.
Often local data agree on overlaps but still do not come from one global object.
Sheaf cohomology counts such obstructions.
For example, on the unit circle in $\C$, a continuous square root of $z$ can be chosen on each small arc.
But these choices cannot be made to fit together around the whole circle.
This obstruction is a nonzero class in $H^1$ of the circle with coefficients in $\Z/2$.