A Tannakian context for Galois theory (Q1935192)

Roughly speaking, categorical Galois theory and Tannaka theory are both about recognizing categories of ``group actions''. However, Tannaka theory is closely tied to an abelian context (with groups being algebraic and represented by Hopf algebras), whereas categorical Galois theory is generally more about actions on sets (with the relevant sort of groups being topological or localic). The present paper unifies the two in a satisfying way. First of all, the authors invoke the well-known fact that most topos-theoretic notions can be formulated using the category of internal relations in the topos. Secondly, they observe that the category of sets and relations can be identified with the full subcategory of the category of suplattices spanned by powersets. As was first pointed out by \textit{A. Joyal} and \textit{M. Tierney} [Mem. Am. Math. Soc. 309, 71 p. (1984; Zbl 0541.18002)], suplattices behave very much like abelian groups, with locales being identifiable with certain rings therein; thus (an abstract version of) Tannakian theory can be applied. Roughly speaking, where traditional Tannaka theory invokes the duality between algebraic spaces and commutative rings, we instead invoke the analogous duality between locales and their frames of opens. The paper's main result is that when this correspondence is applied to a pointed topos, the localic group of automorphisms assigned by Galois theory corresponds to the Hopf algebra in suplattices assigned by Tannaka theory, and one comparison functor is an equivalence if and only if the other is. This paper does suffer a slight defect in that Sections 3 and 4 are difficult to read as-is, since they consist of studying a large number of unmotivated abstract conditions with unevocative names. However, skipping forward to Section 5 should provide the missing motivation. Also, although the authors don't mention it, their ``elevator calculus'' appears to be a variant of what is now commonly called ``string diagram'' calculus. However, these minor issues do not detract from the paper's value.