A Construction of a Quotient Tensor Category

Let f : G → A be a surjective homomorphism of transitive groupoid schemes and let L denote the kernel of f . The exact sequence of groupoid schemes 1 → L → G → A → 1 induces a sequence of functors between the categories of finite representations of these groupoid schemes Repf (A) → Repf (G) → Repf (L). We show that the category Repf (L) is a quotient category of Repf (G) by Repf (A) in an appropriate sense. We also generalize this setting to the framework where the tensor categories are not necessarily Tannaka categories (i.e. not of the form Repf (G) for some groupoid scheme G), where we show under certain assumption the uniqueness of the quotient tensor category.