Sato Grassmannians for generalized Tate spaces

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.

💡 Research Summary

The paper presents a systematic generalisation of Sato Grassmannians, originally defined for locally linearly compact topological vector spaces (Tate spaces), to the setting of “locally compact objects” in an arbitrary exact category A. The authors introduce the ind‑pro limit category lim A, whose objects are simultaneously filtered colimits and cofiltered limits of objects of A, and they show that this construction captures the essential features of Tate spaces while being applicable to any exact category.

For an object X in lim A, a “lattice” is defined as an admissible subobject that is both a monomorphism and an epimorphism in the exact structure; the collection of all lattices forms a partially ordered set Gr(X), which the authors call the Sato Grassmannian of X. This Grassmannian inherits the familiar operations of sum and intersection, and these operations respect admissible short exact sequences, mirroring the classical theory.

Using the lattice structure, the authors construct two torsors associated with X: the dimensional torsor Dim(X), a ℤ‑torsor measuring the relative “size” of lattices, and the determinantal gerbe Det(X), a ℂ×‑gerbe encoding the determinant line bundles attached to lattices. They then combine these into a single object D(X), a 2‑groupoid (or determinantal torsor) that simultaneously records both dimension and determinant data and is independent of the choice of lattice.

A central technical contribution is the introduction of “partially abelian exact” categories. An exact category A is partially abelian exact if every admissible monomorphism and epimorphism admits a kernel and cokernel, and if these kernels and cokernels satisfy the usual abelian exchange laws. Under this hypothesis, the authors prove multiplicativity of both Dim and Det with respect to admissible short exact sequences: \