#
Tannaka theory over sup-lattices and descent for topoi

##
Eduardo J. Dubuc and Martin Szyld

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.

http://www.tac.mta.ca/tac/volumes/31/31/31-31.pdf

TAC Home