Locally constant sheaf

In algebraic topology, a locally constant sheaf on a topological space X is a sheaf ${\displaystyle {\mathcal {F}}}$ on X such that for each x in X, there is an open neighborhood U of x such that the restriction ${\displaystyle {\mathcal {F}}|_{U}}$ 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 ${\displaystyle X=\mathbb {C} }$, ${\displaystyle {\mathcal {O}}_{X}}$ be the sheaf of holomorphic functions on X and ${\displaystyle P:{\mathcal {O}}_{X}\to {\mathcal {O}}_{X}}$ given by ${\displaystyle P=z{\partial \over \partial z}-{1 \over 2}}$. Then the kernel of P is a locally constant sheaf on ${\displaystyle X-\{0\}}$ but not constant there (since it has no nonzero global section).^[1]

If ${\displaystyle {\mathcal {F}}}$ is a locally constant sheaf of sets on a space X, then each path ${\displaystyle p:[0,1]\to X}$ in X determines a bijection ${\displaystyle {\mathcal {F}}_{p(0)}{\overset {\sim }{\to }}{\mathcal {F}}_{p(1)}.}$ Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor

${\displaystyle \Pi _{1}X\to \mathbf {Set} ,\,x\mapsto {\mathcal {F}}_{x}}$

where ${\displaystyle \Pi _{1}X}$ 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 ${\displaystyle \Pi _{1}X\to \mathbf {Set} }$ is of the above form; i.e., the functor category ${\displaystyle \mathbf {Fct} (\Pi _{1}X,\mathbf {Set} )}$ 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]