In algebraic topology, a locally constant sheaf on a topological space X is a sheaf
When X is a stratified space, a constructible sheaf is roughly a sheaf that is locally constant on each member of the stratification.
A basic example is the orientation sheaf on a manifold since each point of the manifold admits an orientable open neighborhood (while the manifold itself may not be orientable).
Then the kernel of P is a locally constant sheaf on
but not constant there (since it has no nonzero global section).
is a locally constant sheaf of sets on a space X, then each path
Moreover, two homotopic paths determine the same bijection.
is the fundamental groupoid of X: the category whose objects are points of X and whose morphisms are homotopy classes of paths.
Moreover, if X is path-connected, locally path-connected and semi-locally simply connected (so X has a universal cover), then every functor
is of the above form; i.e., the functor category
is equivalent to the category of locally constant sheaves on X.
If X is locally connected, the adjunction between the category of presheaves and bundles restricts to an equivalence between the category of locally constant sheaves and the category of covering spaces of X.