A Tannakian Context for Galois

Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hold, and where the unit of the tensor product is always an object of finite presentation, which is not the case in our context.

💡 Research Summary

The paper establishes a precise categorical bridge between Grothendieck‑style Galois theory of atomic topoi and neutral Tannakian theory. Starting from a pointed topos (E) equipped with a set‑valued point (F:E\to\mathbf{Set}), the authors consider the inverse image functor (F) and its associated relation functor (T=\operatorname{Rel}(F)). The key observation is that the category of relations (\operatorname{Rel}(E)) can be identified with the full subcategory (\mathbf{Sup}_0) of the sup‑lattice category (\mathbf{Sup}) consisting of power‑set objects (\ell X). This identification supplies the monoidal base ((\mathbf{V},\mathbf{V}_0)=(\mathbf{Sup},\mathbf{Sup}_0)) required for a neutral Tannakian context.

Within this framework a localic group (G=\operatorname{Aut}(F)) (the automorphism group of the point) is viewed as an idempotent Hopf algebra in (\mathbf{Sup}). Dually, the endomorphism Hopf algebra (H=\operatorname{End}^\vee(T)) of the functor (T) is constructed inside (\mathbf{Sup}_0). Two central theorems are proved:

1. Theorem 4.6: For any localic group (G) there is an equivalence of monoidal categories \