  We categorify various Fock space representations on the algebra of symmetric
functions via the category of polynomial functors. In a prequel, we used
polynomial functors to categorify the Fock space representations of type A
affine Lie algebras. In the current work we continue the study of polynomial
functors from the point of view of higher representation theory. Firstly, we
categorify the Fock space representation of the Heisenberg algebra on the
category of polynomial functors. Secondly, we construct commuting actions of
the affine Lie algebra and the level p action of the Heisenberg algebra on the
derived category of polynomial functors over a field of characteristic p > 0,
thus weakly categorifying the Fock space representation of $\hat{gl}_p$.
Moreover, we study the relationship between these categorifications and
Schur-Weyl duality. The duality is formulated as a functor from the category of
polynomial functors to the category of linear species. The category of linear
species is known to carry actions of the Kac-Moody algebra and the Heisenberg
algebra. We prove that Schur-Weyl duality is a morphism of these
categorification structures.
