The category of representations of a Lie groupoid

A Lie groupoid is a groupoid with additional smooth manifold structures on the object set and the morphism set that makes various maps arise from the groupoid structure smooth. In this paper, we describe the properties of the category Rep([Formula: see text]) whose objects are the classical representations of the Lie groupoid [Formula: see text] and morphisms are the base preserving vector bundle morphisms that respects the representations. It is shown that this category is an additive category which is monoidal with subobjects. Also we have discussed the kernel and cokernel for certain types of morphisms.