Additive Grothendieck pretopologies and presentations of tensor categories

TL;DR

This paper introduces additive Grothendieck pretopologies for preadditive categories, enabling new ways to present tensor categories and analyze their universal properties and sheaf categories.

Contribution

It defines additive pretopologies analogous to Grothendieck pretopologies, facilitating the study of tensor categories and their presentations.

Findings

Defined additive Grothendieck pretopologies for preadditive categories

Characterized how these pretopologies influence sheaf categories

Showed applications to presenting tensor categories with universal properties

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.