Generic bundles over a localic category

TL;DR

This paper constructs classifying localic categories and groupoids for various logical bundles, showing they classify more theories than toposes and providing concrete, constructive methods.

Contribution

It introduces a concrete, constructive approach to classifying localic categories and groupoids for logical bundles, extending the scope beyond toposes.

Findings

Classifying localic categories and groupoids for bundles with logical structure.

Recovery of localic groupoids classifying geometric theories with stronger universal properties.

Construction of these categories using generalized frame presentations and a pointfree Alexandroff--Hausdorff theorem.

Abstract

In this paper we construct classifying localic categories and groupoids for various bundles equipped with logical structure. When these bundles are local homeomorphisms, we recover the localic groupoids that classify geometric theories, demonstrating that these groupoids satisfy a stronger universal property than that of their corresponding classifying toposes. We also prove a dual result that there exist classifying localic categories and groupoids for proper separated bundles satisfying a dual geometric theory. Thus, localic groupoids classify strictly more kinds of logical theories than toposes. Our approach provides a concrete construction of the localic categories and the generic bundles involved in terms of generalised frame presentations. To accommodate our approach, we prove en passant a constructive, pointfree version of the Alexandroff--Hausdorff theorem and that internal functors that are fully faithful and effective descent morphisms on objects induce equivalences between the categories of discrete opfibrations over the source and target categories.