Tannaka theory over sup-lattices and descent for topoi

SZYLD, MARTÍN; DUBUC, EDUARDO J

THEORY AND APPLICATIONS OF CATEGORIES

Mount Allison University Library

Año: 2016 vol. 31 p. 852 - 906

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 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).