Topological semiinfinite tensor (super)modules

We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras gl(V⊕ΠV) and osp(V⊕ΠV) associated with a Tate space V. Here V⊕ΠV is a Z/2Z-graded topological vector space whose even and odd parts are isomorphic to V. We discuss the purely even case first, by introducing monoidal categories Tˆgl(V), Tˆo(V) and Tˆsp(V), and show that these categories are anti-equivalent to respective previously studied categories Tgl(V), To(V), Tsp(V). These latter categories have certain universality properties as monoidal categories, which consequently carry over to Tˆgl(V), Tˆo(V) and Tˆsp(V). Moreover, the categories To(V) and Tsp(V) are known to be equivalent, and this implies the equivalence of the categories Tˆo(V) and Tˆsp(V). After introducing a supersymmetric setting, we establish the equivalence of the category Tˆgl(V) with the category Tˆgl(V⊕ΠV), and the equivalence of both categories Tˆo(V) and Tˆsp(V) with Tˆosp(V⊕ΠV).