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Local systems

Generally speaking, representations may serve to exhibit features of a group by its actions on other mathematical objects, often vector spaces. Representations of the fundamental group have a very geometric significance: any local system (i.e., a sheaf F {\displaystyle {\mathcal {F}}} on X with the property that locally in a sufficiently small neighborhood U of any point on X, the restriction of F is a constant sheaf of the form F | U = Q n {\displaystyle {\mathcal {F}}|_{U}=\mathbb {Q} ^{n}}) gives rise to the so-called monodromy representation, a representation of the fundamental group on an n - dimensional Q {\displaystyle \mathbb {Q} } -vector space. Conversely, any such representation on a path-connected space X arises in this manner. This equivalence of categories between representations of π 1 (X) {\displaystyle \pi _{1}(X)} and local systems is used, for example, in the study of differential equations, such as the Knizhnik–Zamolodchikov equations.