Locally constant sheaf

In algebraic topology, a locally constant sheaf on a topological space "X" is a sheaf formula_1 on "X" such that for each "x" in "X", there is an open neighborhood "U" of "x" such that the restriction formula_2 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 formula_3, formula_4 be the sheaf of holomorphic functions on "X" and formula_5 given by formula_6. Then the kernel of "P" is a locally constant sheaf on formula_7 but not constant there (since it has no nonzero global section).
If formula_1 is a locally constant sheaf of sets on a space "X", then each path formula_9 in "X" determines a bijection formula_10 Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor
where formula_12 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 formula_13 is of the above form; i.e., the functor category formula_14 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".