Locally constant sheaf

In algebraic topology, a locally constant sheaf on a topological space X is a sheaf $$\mathcal{F}$$ on X such that for each x in X, there is an open neighborhood U of x such that the restriction $$\mathcal{F}|_U$$ 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 (while the manifold itself may not be orientable).

For another example, let $$X = \mathbb{C}$$, $$\mathcal{O}_X$$ be the sheaf of holomorphic functions on X and $$P: \mathcal{O}_X \to \mathcal{O}_X$$ given by $$P = z {\partial \over \partial z} - {1 \over 2}$$. Then the kernel of P is a locally constant sheaf on $$X - \{0\}$$ but not constant there (since it has no nonzero global section).

If $$\mathcal{F}$$ is a locally constant sheaf of sets on a space X, then each path $$p: [0, 1] \to X$$ in X determines a bijection $$\mathcal{F}_{p(0)} \overset{\sim}\to \mathcal{F}_{p(1)}.$$ Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor
 * $$\Pi_1 X \to \mathbf{Set}, \, x \mapsto \mathcal{F}_x$$

where $$\Pi_1 X$$ 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 $$\Pi_1 X \to \mathbf{Set}$$ is of the above form; i.e., the functor category $$\mathbf{Fct}(\Pi_1 X, \mathbf{Set})$$ 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.