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Concept
Local system
Summary
In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943.
The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold.
Let X be a topological space. A local system (of abelian groups/modules/...) on X is a locally constant sheaf (of abelian groups/modules...) on X. In other words, a sheaf is a local system if every point has an open neighborhood such that the restricted sheaf is isomorphic to the sheafification of some constant presheaf.
If X is path-connected, a local system of abelian groups has the same stalk L at every point. There is a bijective correspondence between local systems on X and group homomorphisms
and similarly for local systems of modules. The map giving the local system is called the monodromy representation of .
This shows that (for X path-connected) a local system is precisely a sheaf whose pullback to the universal cover of X is a constant sheaf.
This correspondence can be upgraded to an equivalence of categories between the category of local systems of abelian groups on X and the category of abelian groups endowed with an action of (equivalently, -modules).
A stronger nonequivalent definition that works for non-connected X is: the following: a local system is a covariant functor
from the fundamental groupoid of to the category of modules over a commutative ring , where typically . This is equivalently the data of an assignment to every point a module along with a group representation such that the various are compatible with change of basepoint and the induced map on fundamental groups.
Constant sheaves such as . This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:
Let .
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