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 q : shB --> E in Galois theory and a Tannakian recognition theorem over sl for the sl-functor Rel (q^*) : Rel(E) --> Rel(shB) \cong (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 \cong O(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 = 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.
Keywords: Tannaka, Galois, Sup-lattice, Locale, Topos
2010 MSC: Primary: 18F99. Secondary: 14L99, 18B25
Theory and Applications of Categories, Vol. 31, 2016, No. 31, pp 852-906.
Published 2016-10-06.
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