Locally constant sheaf
In algebraic topology, a locally constant sheaf on a topological space X is a sheaf on X such that for each x in X, there is an open neighborhood U of x such that the restriction is a constant sheaf on U. It is also called a local system. 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
For another example, let, be the sheaf of holomorphic functions on X and given by. Then the kernel of P is a locally constant sheaf on but not constant there.
If is a locally constant sheaf of sets on a space X, then each path in X determines a bijection Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor
where 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, then every functor is of the above form; i.e., the functor category is equivalent to the category of locally constant sheaves on X.
The category of locally constant sheaves of sets on a space X is equivalent to the category of covering spaces of X.
