We categorify various Fock space representations on the algebra of symmetric
functions via the category P of strict polynomial functors of finite
degree. More precisely, we use the category P to categorify the Fock space
representations of type A affine Lie algebra and Heisenberg algebras, and we
also categorify the commutativity of the action of the affine Lie algebras
and the level p action of Heisenberg algebras on the Fock space. Moreover,
we study the relationship between these categorifications and SchurWeyl
duality. The duality is formulated as a functor from the category P to the
category of linear species, and we prove that SchurWeyl duality is a
morphism of these categorification structures.
In the end, we propose a definition of Hopf
category, and we show that the category of polynomial functors is naturally
endowed with a Hopf category structure.
