The theory of polynomial functors allows one to make sense of the stable polynomial representation theory of the general linear group over a field of characteristic 0. It also has a good notion of specialization, so that calculations done in the "infinite limit" can be used to get information in the usual finite-dimensional siutation. (At the combinatorial level this is understood as the theory of symmetric functions vs. symmetric polynomials.) Such a theory is less well-understood in the other classical groups, but the analogous category has been introduced by Olshanskii and Penkov-Strykas. However, the specialization from the infinite case to the finite case (which is needed for applications) was not previously understood. This talk will explain joint work with Andrew Snowden and Jerzy Weyman where such a specialization is constructed and its properties analyzed. Time permitting, I will explain how this is intricately connected with the invariant theory and commutative algebra surrounding determinantal varieties. I will not assume prior knowledge of polynomial functors and representation theory of classical groups.