Differential Tannakian Formalism

1.1. Definition. Let C be an abelian category. The derivative D(C) of C is defined as follows: The objects are exact sequences 0 −→ X0 iX −−→ X1 πX −−→ X0 −→ 0 of C, and the morphisms from such an object are morphisms of exact sequences whose two X parts coincide. The category D(C) is again abelian. An exact functor F : C1 −→ C2 gives rise to an induced (exact) functor D(F ) : D(C1) −→ D(C2). We denote by Πi (i = 0, 1) the functors from D(C) to C assigning Xi to 0 −→ X0 iX −−→ X1 πX −−→ X0 −→ 0 (thus there is an exact sequence 0 −→ Π0 iΠ −→ Π1 πΠ −−→ Π0 −→ 0.) Πi(X) is also abbreviated as Xi, and X is said to be over X0 (and similarly for morphisms.)