# 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).^{[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*.

The category of locally constant sheaves of sets on a space *X* is equivalent to the category of covering spaces of *X*.^{[citation needed]}

## References[edit]

- Kashiwara, Masaki; Schapira, Pierre (2002),
*Sheaves on Manifolds*, Berlin: Springer, ISBN 3540518614 - § A.1. of J. Lurie, Higher Algebra, last updated May 2016.

## External links[edit]

- https://ncatlab.org/nlab/show/locally+constant+sheaf
- https://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html (recommended)

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**^**Kashiwara–Schapira, Example 2.9.14.