# Correspondence functors and finiteness conditions

## TL;DR

This paper studies the representation theory of finite sets through correspondence functors, revealing conditions under which these functors are finitely generated and have finite length, with implications for their structural properties.

## Contribution

It characterizes when correspondence functors are finitely generated and have finite length, providing new insights into their structural and finiteness properties.

## Key findings

- Finitely generated correspondence functors grow exponentially in dimension.
- Such functors have finite length.
- Subfunctors of finitely generated functors are also finitely generated when the ring is noetherian.

## Abstract

We investigate the representation theory of finite sets. The correspondence functors are the functors from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. They have various specific properties which do not hold for other types of func-tors. In particular, if k is a field and if F is a correspondence functor, then F is finitely generated if and only if the dimension of F (X) grows exponentially in terms of the cardinality of the finite set X. Moreover, in such a case, F has actually finite length. Also, if k is noetherian, then any subfunctor of a finitely generated functor is finitely generated.