Paper detail

Tannaka Theory for Topos

We consider locales $B$ as algebras in the tensor category $s\ell$ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections $sh(B) \stackrel{q}{\longrightarrow} \mathcal{E}$ in Galois theory [An extension of the Galois Theory of Grothendieck, AMS Memoirs 151] and a Tannakian recognition theorem over $s\ell$ for the $s\ell$-functor $Rel(E) \stackrel{Rel(q^*)}{\longrightarrow} Rel(sh(B)) \cong (B$-$Mod)_0$ into the $s\ell$-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, $L \cong \mathcal{O}(G)$. On the other hand, we show that the $s\ell$-category of relations of the classifying topos of any localic groupoid $G$, is equivalent to the $s\ell$-category of $L$-comodules with discrete subjacent $B$-module, where $L = \mathcal{O}(G)$. We are forced to work over an arbitrary base topos because, contrary to the neutral case developed over Sets in [A Tannakian Context for Galois Theory, Advances in Mathematics 234], here change of base techniques are unavoidable.