Tannaka theory over sup-lattices and descent for topoi

DUBUC, EDUARDO J.; MARTIN SZYLD

THEORY AND APPLICATIONS OF CATEGORIES

Mount Allison University

Año: 2016

1201-561X

We consider locales B as algebras in the tensor category sℓ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections sh(B)⟶q in Galois theory [An extension of the Galois Theory of Grothendieck, AMS Memoirs 151] and a Tannakian recognition theorem over sℓ for the sℓ-functor Rel(E)⟶Rel(q∗)Rel(sh(B))≅(B-Mod)0 into the sℓ-category of discrete B-modules. Thus, a new Tannaka recognition theorem is obtained, essentially different from those known so far. This equivalence follows from two independent results. We develop an explicit construction of the localic groupoid G associated by Joyal-Tierney to q, and do an exhaustive comparison with the Deligne Tannakian construction of the Hopf algebroid L associated to Rel(q∗), and show they are isomorphic, that is, L≅(G). On the other hand, we show that the sℓ-category of relations of the classifying topos of any localic groupoid G, is equivalent to the sℓ-category of L-comodules with discrete subjacent B-module, where L=(G). We are forced to work over an arbitrary base topos because, contrary to the neutral case developed over Sets in [A Tannakian Context for Galois Theory, Advances in Mathematics 234], here change of base techniques are unavoidable.