Tannaka Theory for Topos

TL;DR

This paper develops a new Tannakian recognition theorem for locales and localic groupoids, establishing an equivalence with Galois theory and comparing it to Deligne's Hopf algebroid construction, broadening the scope of Tannakian duality.

Contribution

It introduces a novel Tannakian recognition theorem for localic groupoids and relates it to Galois theory, extending Tannakian duality beyond the classical setting.

Findings

Constructed the localic groupoid G associated to q

Proved L is isomorphic to the Hopf algebroid of G

Established equivalence between relations of the classifying topos and L-comodules

Abstract

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.