algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

, a locally constant sheaf on a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

''X'' is a

sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * '' The Sheaf'', a student-run newspaper s ...

\mathcal

on ''X'' such that for each ''x'' in ''X'', there is an open neighborhood ''U'' of ''x'' such that the restriction

\mathcal, _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

\mathcal_X

be the sheaf of holomorphic functions on ''X'' and

P: \mathcal_X \to \mathcal_X

given by

P = z  -

. Then the kernel of ''P'' is a locally constant sheaf on

X - \

but not constant there (since it has no nonzero global section). If

\mathcal

is a locally constant sheaf of sets on a space ''X'', then each path

p:

, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

\to X in ''X'' determines a bijection

\mathcal_ \overset\to \mathcal_.

Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor :

\Pi_1 X \to \mathbf, \, x \mapsto \mathcal_x

where

\Pi_1 X

is the

fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a ...

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

is of the above form; i.e., the functor category

\mathbf(\Pi_1 X, \mathbf)

is equivalent to the category of locally constant sheaves on ''X''. If ''X'' is locally connected, the adjunction between the category of

presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...

and bundles restricts to an equivalence between the category of locally constant sheaves and the category of covering spaces of ''X''.

References

* *

External links

* *https://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html (recommended) Algebraic topology Topological spaces {{topology-stub