Cartesian modules over representations of small categories

TL;DR

This paper introduces the concept of cartesian modules over pseudofunctors from small categories to preadditive categories, generalizing classical sheaf constructions and extending representation theory and derived categories in algebraic geometry.

Contribution

It defines cartesian modules over pseudofunctors, generalizing quasi-coherent sheaves and extending Crawley-Boevey's Representation Theorem in a broader categorical framework.

Findings

Provides a new framework for modules over pseudofunctors

Extends classical sheaf and representation theories

Relates to and extends pure derived categories of schemes

Abstract

We introduce the new concept of cartesian module over a pseudofunctor $R$ from a small category to the category of small preadditive categories. Already the case when $R$ is a (strict) functor taking values in the category of commutative rings is sufficient to cover the classical construction of quasi-coherent sheaves of modules over a scheme. On the other hand, our general setting allows for a good theory of contravariant additive locally flat functors, providing a geometrically meaningful extension of Crawley-Boevey's Representation Theorem. As an application, we relate and extend some previous constructions of the pure derived category of a scheme.