arXiv subjects

Joshua L. Wrigley

Publications and source records attributed to Joshua L. Wrigley.

5 recordsLinked to original sources

arXiv2026

Representation of abstract pseudogroups

The Wagner-Preston theorem states that every inverse semigroup can be embedded into an inverse semigroup of partial bijections on some set. However, this embedding does not respect lattice operations in the natural ordering. In this paper, we show that an abstract pseudogroup can be faithfully represented by partial bijections on a set if and only if its frame of idempotents is spatial. More generally, we show that any abstract pseudogroup can be faithfully represented by partial homeomorphisms on a locale.

math.RA↗

arXiv2026

The points of canonical extensions of doctrines

We analyse the space of points of the canonical extension of a coherent doctrine. We first give a full characterisation of doctrine morphisms that are extensible, and relate it to the existing notion of p-model of a coherent category. Through this characterisation, the extensible morphisms are shown to be exactly those which are ω-saturated in the sense of coherent first-order logic. Next, we answer the question: when does a presheaf of models fully describe the canonical extension? We prove a characterisation theorem via two conditions, which are again natural from the perspective of coherent logic, namely, homogeneity and the realisation of all prime types in a strict sense. The characterisation theorem allows us to deduce a reconstruction result for any coherent theory with the property that all prime types can be realised in a countable, saturated model. For instance, ω-stable coherent theories always have this property. We conclude by explaining how our results can be interpreted topos-theoretically, by relating them to the classifying topos and to the topos of types.

math.CT↗

arXiv2026

Generic bundles over a localic category

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.

math.CT↗

arXiv2025

Existential completions and Herbrand's theorem

Recently, Abbadini and Guffanti gave an algebraic proof of Herbrand's theorem using a completion for Lawvere doctrines that freely adds existential and universal quantifiers. A more direct argument can be given by only completing with respect to existential quantifiers. We construct the free existential completion on a presheaf of distributive lattices, and deduce Herbrand's theorem for coherent logic from the explicit description. We also discuss the cases involving presheaves of meet-semilattices, due to Trotta, and presheaves of frames.

math.LO↗

arXiv2024

The representing localic groupoid for a geometric theory

We give an expository, and hopefully approachable, account of the Joyal-Tierney result that every topos can be represented as a topos of sheaves on a localic groupoid. We give an explicit presentation of a representing localic groupoid for the classifying topos of a given geometric theory and discuss links with the topological groupoids of Forssell.

math.CT↗