Inductive systems of the symmetric group, polynomial functors and tensor categories

We initiate the systematic study of modular representations of symmetric groups that arise via the braiding in (symmetric) tensor categories over fields of positive characteristic. We determine what representations appear for certain examples of tensor categories, develop general principles and demonstrate how this question connects with the ongoing study of the structure theory of tensor categories. We also formalise a theory of polynomial functors as functors that act coherently on all tensor categories. We conclude that the classification of such functors is a different way of posing the above question of which representations of symmetric groups appear. Finally, we extend the classical notion of strict polynomial functors from the category of (super) vector spaces to arbitrary tensor categories, and show that this idea is also a different packaging of the same information.