Functors on the category of finite sets revisited

We study the structure of the category of representations of $\mathbf{FA}$, the category of finite sets and all maps, mostly working over a field of characteristic zero. This category is not semi-simple and exhibits interesting features. We first construct the simple representations, recovering the classification given by Wiltshire-Gordon. The construction given here also yields explicit descriptions of the indecomposable projectives. These results are used to give a convenient set of projective generators of the category of representations of $\mathbf{FA}$ and hence a Morita equivalence result. This is used to explain how to calculate the multiplicities of the composition factors of an arbitrary object, based only on its underlying $\mathbf{FB}$-representation, where $\mathbf{FB}$ is the category of finite sets and bijections. This is 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 (the significance of which can be shown by relative nonhomogeneous Koszul duality theory).