Additive Grothendieck Pretopologies and Presentations of Tensor Categories

We study how tensor categories can be presented in terms of rigid monoidal categories and Grothendieck topologies and show that such presentations lead to strong universal properties. As the main tool in this study, we define a notion on preadditive categories which plays a role similar to (a generalisation of) the notion of a Grothendieck pretopology on an unenriched category. Each such additive pretopology defines an additive Grothendieck topology and suffices to define the sheaf category. This new notion also allows us to study the noetherian and subcanonical nature of additive topologies, to describe easily the join of a family of additive topologies and to identify useful universal properties of the sheaf category.