Polynomial functors and categorifications

The theory of symmetric functions is very classical and fundamental in mathematics. The space B of symmetric functions admits many symmetries. For example, it is a Hopf algebra. There also exists representation structures of type A affine Lie algebra and Heisenberg algebra on the space B.
Usually they are called Fock space representation.
The theory of polynomial functors is also very fundamental in mathematics. It is closely related to general linear groups. The category P of polynomial functors is a natural categorification of the space B. There exists natural categorifications of Fock space representations of affine Lie algebra and Heisenberg algebra. Moreover, the category P admits a natural Hopf category structure.
Schur-Weyl duality relates polynomial representation of general linear groups and representation of symmetric groups. The Schur-Weyl duality functor can be enriched to be a morphism of various categorifications from the cateogry P to the category of representations of symmetric groups.