Locally constant sheaf

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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 $X=\mathbb {C}$ , ${\mathcal {O}}_{X}$ 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).

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.

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