Tannaka theory over sup-lattices and descent for topoi

Eduardo J. Dubuc ; Martin Szyld

Theory and applications of categories, Tome 31 (2016), pp. 852-906

Résumé

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.

Détail

Publié le : 2016-10-06

Classification : Primary: 18F99. Secondary: 14L99, 18B25
Keywords: Tannaka, Galois, Sup-lattice, Locale, Topos