Simple and projective correspondence functors

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