  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 and unify their treatment in
the determinantal torsor D(X). We then introduce a class of exact categories,
that we call partially abelian exact, and prove that if A is partially abelian
exact, Dim(X) and Det(X) are multiplicative in admissible short exact
sequences. When A is the category of finite dimensional vector spaces on a
field k, we recover the case of the dimensional torsor and of the determinantal
gerbe of a Tate space, as defined by Kapranov and reformulate its properties in
terms of the Waldhausen space S(A) of the exact category A. The advantage of
this approach is that it allows to define formally in the same way the
Grassmannians of the iterated categories lim^nA. We then prove that the
category of Tate spaces is partially abelian exact, which allows us to extend
the results on Dim and Det already known for Tate spaces to 2-Tate spaces, such
as the multiplicativity of Dim and Det for 2-Tate spaces, as considered by
Arkhipov-Kremnizer and Frenkel-Zhu.
