Tannaka Theory over Sup-Lattices

TL;DR

This thesis constructs a tannakian framework over the category of sup-lattices associated with any Grothendieck topos, extending Galois and Tannaka theories and establishing their fundamental equivalence.

Contribution

It introduces a novel tannakian context over sup-lattices for arbitrary topoi, generalizing existing Galois and Tannaka theories and proving their equivalence.

Findings

Established a tannakian context over sup-lattices for any Grothendieck topos.

Proved the equivalence of Galois and Tannaka recognition theorems.

Derived a new recognition-type tannakian theorem based on relations of a topos.

Abstract

The main result of this thesis is the construction of a tannakian context over the category of sup-lattices, associated with an arbitrary Grothendieck topos, and the attainment of new results in tannakian representation theory from it. Although many results were obtained and published historically linking Galois and Tannaka theory (see our introduction), these are different and less general since they assume the existence of Galois closures and work on Galois topos rather than on arbitrary topos. Instead we, when talking about Galois theory, mean the extension to arbitrary topos of Joyal and Tierney, critical to get the results of this thesis. The tannakian context associated with a Grothendieck topos is obtained through the process of taking relations of its localic cover. Then, through an investigation and exhaustive comparison of the constructions of the Galois and Tannaka theories, we prove the equivalence of their fundamental recognition theorems (see section 8). Since the (bi)categories of relations of a Grothendieck topos were characterized by Carboni and Walters, a new recognition-type tannakian theorem (theorem 8.12) is obtained, essentially different from those known so far.