On the relation between Grothendieck-Galois and Deligne-Tannaka theories

DUBUC, EDUARDO J.

Como

Congreso; Toposes in Como 2018; 2018

Universita dell'Insubria

Strong similarities have been long observed between dierent ver-sions of Galois and Tannaka representation theories. However, thesesimilarities are just of form, and don't allow to transfer any result fromone theory to another, in particular Galois theory and Tannaka the-ory (over vector spaces) remain independent. As a preamble, we willbriey recall the Grothendieck-Galois and the neutral Deligne-Tannakaversions of these theories.Observing that the category of relations of a Grothendieck toposis a category enriched over sup-lattices, we relate the Galois contextto the Tannakian context over the tensor category s` of sup-lattices.We develop the case of the (localic) group G of automorphisms of aSet-valued ber functor F, on the Galois side, and the Hopf algebra Hof endomorphism of a s`-valued ber functor T, on the Tannaka side.This correspondence is obtained via the category of relations functorRel. We establish an isomorphism between G and H for T = Rel(F),and between the categories of actions of G on a set and the categoryof comodules of G on a free sup-lattice. This yields an equivalencebetween the respective recognition theorems. The general case dealswith localic groupoids and Hopf algebroids, and we obtain similar re-sults. This concerns the Joyal-Tierney generalisation of Galois theory,and it becomes necessary to work over an arbitrary base topos becausechange of base techniques become essential and unavoidable. In thistalk we will concentrate on the neutral case which is within the scopeof a larger audience, and only mention the general case.