Definition
Local system
A locally constant sheaf, equivalently on a suitable space a representation of its fundamental groupoid.
Definition
A local system of -modules on a topological space is a sheaf of -modules that is locally isomorphic to a constant sheaf. More generally, a local system may take values in sets, groups, vector spaces, or another category.
On a path-connected, locally path-connected, and semilocally simply connected space, choosing identifies finite-rank local systems with representations
up to change of basis.
Fundamental-groupoid form
Without choosing a base point, parallel continuation gives a functor from the fundamental groupoid of . This formulation handles disconnected spaces and makes transport along paths intrinsic.
Relation to flat bundles
A finite-rank complex local system determines a vector bundle with a flat connection. Conversely, horizontal sections of a flat bundle form a local system. This analytic correspondence should not be confused with the algebraic de Rham formulation of a -local system.
References
- Alexander Grothendieck, Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics 224, Springer, 1971. DOI.