We generalize the concept of Sato Grassmannians of locally linearly compact
topological vector spaces (Tate spaces) to the category limA of the "locally
compact objects" of an exact category A, and study some of their properties.
This allows us to generalize the Kapranov dimensional torsor Dim(X) and
determinantal gerbe Det(X) for the objects of limA. We then introduce a class
of exact categories, that we call quasiabelian exact, and prove that if A is
quasiabelian exact, Dim(X) and Det(X) are multiplicative in admissible short
exact sequences.