Sheaf cohomology in topology

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.