# Simple and projective correspondence functors

## TL;DR

This paper characterizes simple projective correspondence functors and explores their occurrence within functors associated with finite lattices, leading to a decomposition result.

## Contribution

It precisely identifies which simple correspondence functors are projective and analyzes their role in lattice-based functors, providing a decomposition.

## Key findings

- Characterization of simple projective correspondence functors
- Decomposition of lattice-associated correspondence functors
- Identification of simple projectives within specific functors

## Abstract

A correspondence functor is a functor from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring. We determine exactly which simple correspondence functors are projective. Moreover, we analyze the occurrence of such simple projective functors inside the correspondence functor $F$ associated with a finite lattice and we deduce a direct sum decomposition of $F$.