ABSTRACT
This chapter introduces the notions of linear representations of categories and groupoids. It first discusses what a linear representation of a quiver is and extends it naturally to define the notion of a linear representation of a category. Using explicit examples of representations of quivers, the chapter shows that irreduciblity and indecomposability are different. The Krull-Schmidt theorem, stating the decomposability of finite-dimensional representations of categories, is proved, and Gabriel’s theorem, characterizing quivers possessing a finite number of indecomposable representations, is stated. The chapter analyzes the relation between linear representations of the isotropy groups and the representations of the groupoid itself. Finally, it proves the Jordan-Hölder theorem for representations of groupoids and shows that any finite-dimensional linear representation of a finite groupoid is completely reducible.
