Generalized correspondence functors

Abstract

Abstract A generalized correspondence functor is a functor from the category of finite sets and T-generalized correspondences to the category of all k-modules, where T is a finite distributive lattice and k a commutative ring. We parametrize simple generalized correspondence functors using the notions of T-module and presheaf of posets. As an application, we prove finiteness and stabilization results. In particular, when k is a field, any finitely generated correspondence functor has finite length, and when k is noetherian, any subfunctor of a finitely generated functor is finitely generated.