We define a faithful linear monoidal functor from the partition category, and hence from Deligne's category Rep(S_t), to the additive Karoubi envelope of the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions. We then generalize the above results to any group G, the case where G is the trivial group corresponding to the case mentioned above. Thus, to every group G we associate a linear monoidal category Par(G) that we call a group partition category. We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of Par(G) into the group Heisenberg category associated to G. This embedding intertwines the natural actions of both categories on modules for wreath products of G. Finally, we prove that the additive Karoubi envelope of Par(G) is equivalent to a wreath product interpolating category introduced by Knop, thereby giving a simple concrete description of that category.

We define the affinization of an arbitrary monoidal category C, corresponding to the category of C-diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to C. The affinization formalizes and unifies many constructions appearing in the literature. In particular, we describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants. When C is rigid, its affinization is isomorphic to its horizontal trace, although the two definitions look quite different. In general, the affinization and the horizontal trace are not isomorphic. We then use the affinization to show our main result, which is an explicit isomorphism between the central charge k reduction of the universal central extension of the elliptic Hall algebra and the trace, or zeroth Hochschild homology, of the quantum Heisenberg category of central charge k. We use this isomorphism to construct large families of representations of the universal extension of the elliptic Hall algebra.