There are four series of classical Lie superalgebras: sl, osp, P and Q. In this talk I consider the direct limits of these superalgebras and study their representations in the tensor algebra generated by the standard and costandard representation.

We will see that complications related to the lack of complete reducibility for finite-dimensional superalgebras disappear at infinity. I define an abelian category of tensor modules for those superalgebras and discuss its properties. In particular, we will see that all categories in question are Koszul and extensions between simple modules can be described in terms of Littlewood--Richardon coefficients. As an example, we interpret Howe duality between orthogonal and symplectic groups at infinity in terms of the Lie superalgebra osp.