Model theory and the Tannakian formalism

The aim of this paper is to exhibit the analogy and relationship between two seemingly unrelated theories. On the one hand, the Tannakian formalism, giving a duality theory between affine group schemes (or, more generally, gerbs) and a certain type of categories with additional structure, the Tannakian categories. On the other hand, a general notion of internality in model theory, valid for an arbitrary first order theory, that gives rise to a definable Galois group. The analogy is made precise by deriving (a weak version of) the fundamental theorem of the Tannakian duality (Theorem 2.8) using the model theoretic internality.

The Tannakian formalism assigns to a group G over a field k, its category of representations Rep G . In the version we are mainly interested in, due to Saavedra (Saavedra-Rivano [24]), the group is an affine group scheme over a field. A similar approach works with groups in other categories, the first due to Krein and Tannaka, concerned with locally compact topological groups. Another example is provided in Section 4. In the algebraic case, the category is the category of algebraic finitedimensional representations. This is a k-linear category, but the category structure alone is not sufficient to recover the group. One therefore considers the additional structure given by the tensor product. The Tannakian formalism says that G can be recovered from this structure, together with the forgetful functor to the category of vector-spaces. The other half of the theory is a description of the tensor categories that arise as categories of representations: any tensor category satisfying suitable axioms is of the form Rep G , provided it has a “fibre functor” into the category of vector spaces. Our main references for this subject are the first three sections of Deligne and Milne [4] and Deligne [3].

In model theory, internality was discovered by Zilber as a tool to study the structure of strongly minimal structure (Zil ½ ber [27]). Later, Poizat realised (in Poizat [23]) that this notion can be used to treat the Galois theory of differential equations. The definable Galois correspondence outlined in Theorem 1.6 has its origins there. Later, the theory was generalised to larger classes of theories (Hrushovski [10], Hart

and Shami [7], etc.), and applied in various contexts (e.g., Pillay [19] extended the differential Galois theory to arbitrary “D-groups” definable in DCF).

In appendix B of Hrushovski [8], internality was reformulated in a way that holds in an arbitrary theory. One is interested in the group of automorphisms G of a definable set Q over another definable set C. A set Q is internal to another set C if, after extending the base parameters, any element of Q is definable over the elements of C. The idea is that under this condition, Q is close enough to C so that the G has a chance to be definable, but the requirement that a base extension is required prevents it from being trivial. The theorem is that indeed G is the group of points of a (pro-) definable group (see Theorem 1.6). This theory is reformulated again in Hrushovski [9], where the same construction is described as an abstract duality theory between definable groupoids in a theory T , and certain expansions of it, called internal covers. It is this formulation that we use.

The main results of the paper appear in Sections 3 and 4. In Section 3 we apply internality to prove the fundamental result on Tannakian categories (Theorem 2.8). This is done by constructing, for a tensor category C an internal cover T C of ACF k . Models of this theory correspond, roughly, with fibre functors on C. The theory of internality provides a definable group in ACF k , and this is the group corresponding to C. The other parts of the theory follow from the Galois theory, and from the abstract duality theory of Hrushovski [9].

The result we prove is weaker than the original Theorem 2.8 in the following ways. First, 2.8 (1) states that a certain functor is representable by an affine algebraic group, but we only prove that its restriction to the category of fields agrees with such a group (see also Question 0.1.3 below). Second, our proof works only in characteristic 0. Finally, 2.8(2) is covered, in part, by the general model theoretic statement 1.10, but the rest of the proof is only sketched, since it is not significantly different from the proof found in Deligne and Milne [4].

On the other hand, the proof has the advantage that it is simple and more “geometric” than the algebraic one. It also has the advantage that the method is applicable in a more general context. One such application, concerning usual Galois theory, is briefly discussed in Remark 3.14. A more detailed application appears in Section 4, where we define the notion of a differential tensor category, and explain how the same model theoretic approach gives an analogous theorem (Theorem 4.5.5) there (though the method again gives a weaker result, in the same way as in the alg

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