A Tannakian Context for Galois

TL;DR

This paper establishes a Tannakian framework for Galois theory of atomic topoi, demonstrating their fundamental equivalence and extending the Tannakian recognition theorem to a new non-additive context.

Contribution

It constructs a neutral Tannakian setting for atomic topoi Galois theory and proves the fundamental theorems' equivalence, extending Tannakian recognition to non-additive contexts.

Findings

Proves the equivalence of Galois and Tannakian fundamental theorems in this context.

Extends Tannakian recognition theorem to non-additive, atomic topos Galois theory.

Provides a new example where the unit object is not of finite presentation.

Abstract

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.