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 (while the manifold itself may not be orientable).

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 (since it has no nonzero global section).[1]

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 (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.[2][3]

References

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  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 (1992). "Sheaves of sets". Sheaves in geometry and logic : a first introduction to topos theory. Ieke Moerdijk. New York: Springer-Verlag. p. 104. ISBN 0-387-97710-4. OCLC 24428855.
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