Additive Grothendieck pretopologies and presentations of tensor categories

Abstract

We define a notion on preadditive categories which plays a role similar to 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 allows us to study the noetherian and subcanonical nature of topologies, to describe easily the meet of a family of topologies and to identify useful universal properties of the sheaf category. As our main application we derive in which ways one can present a tensor category and show that such presentations lead to remarkable universal properties.