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).[1]

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.[2][3]

References

  1. ^ Kashiwara & Schapira 2002, Example 2.9.14.
  2. ^ Szamuely, Tamás (2009). "Fundamental Groups in Topology". Galois Groups and Fundamental Groups. Cambridge University Press. p. 57. ISBN 9780511627064.
  3. ^ Mac Lane, Saunders & Moerdijk, Ieke (1992). [Google Books "Sheaves of sets"]. Sheaves in geometry and logic : a first introduction to topos theory. New York: Springer-Verlag. p. 104. ISBN 0-387-97710-4. OCLC 24428855