Tannaka Theory over Sup-lattices and Descent for Topoi

We consider locales B as algebras in the tensor category s` of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections shB q −→ E in Galois theory and a Tannakian recognition theorem over s` for the s`-functor Rel(E) Rel(q ∗) −→ Rel(shB) ∼= (B-Mod)0 into the s`-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 ∼= O(G). On the other hand, we show that the s`-category of relations of the classifying topos of any localic groupoid G, is equivalent to the s`-category of L-comodules with discrete subjacent B-module, where L = O(G). We are forced to work over an arbitrary base topos because, contrary to the neutral case which can be developed completely over Sets, here change of base techniques are unavoidable.