Functors on the category of finite sets revisited

We study the structure of the category of representations of FA , the category of finite sets and all maps, working over a unital commutative ring. We first exhibit a convenient set of projective generators by exploiting Möbius inversion and establish some general structural results. We then specialize to working over a field of characteristic zero. We construct the simple representations explicitly, together with their projective covers. This recovers the classification of the simples given by Wiltshire-Gordon. Hence we obtain a more efficient set of projective generators of the category of representations of FA and the associated Morita equivalence. This is used to explain how to calculate the multiplicities of the composition factors of an arbitrary object, based only on its underlying FB -representation, where FB is the category of finite sets and bijections. These results are applied to show how to calculate the morphism spaces between projectives in our chosen set of generators, as well as for a closely related family of objects.