Editing Fundamental group (section)

===Local systems===

Generally speaking, [[Group representation|representation]]s may serve to exhibit features of a group by its actions on other mathematical objects, often [[vector space]]s. Representations of the fundamental group have a very geometric significance: any ''[[local system]]'' (i.e., a [[sheaf (mathematics)|sheaf]] <math>\mathcal F</math> 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 <math>\mathcal F|_U = \Q^n</math>) gives rise to the so-called [[monodromy representation]], a representation of the fundamental group on an ''n''-[[dimension (vector space)|dimensional]] <math>\Q</math>-vector space. [[Converse (logic)|Conversely]], any such representation on a path-connected space ''X'' arises in this manner.<ref>{{harvtxt|El Zein|Suciu|Tosun|Uludağ|2010|loc=p. 117, Prop. 1.7}}</ref> This [[equivalence of categories]] between representations of <math>\pi_1(X)</math> and local systems is used, for example, in the study of [[differential equation]]s, such as the [[Knizhnik–Zamolodchikov equations]].