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 , 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).
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.
The category of locally constant sheaves of sets on a space X is equivalent to the category of covering spaces of X.
- Kashiwara, Masaki; Schapira, Pierre (2002), Sheaves on Manifolds, Berlin: Springer, ISBN 3540518614
- § A.1. of J. Lurie, Higher Algebra, last updated May 2016.
- https://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html (recommended)
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- Kashiwara–Schapira, Example 2.9.14.