Representations of quivers on abelian categories and monads on projective varieties

We consider representations of quivers in arbitrary categories and twisted representations of quivers in arbitrary tensor categories. We show that if A is an abelian category, then the category of representations of a quiver in A is also abelian, and that the category of twisted linear representations of a quiver is equivalent to the category of linear (untwisted) representations of a diferent quiver. We conclude by discussing how representations of quivers arise naturally in certain important problems concerning monads ans sheaves on projective varieties.