Jonathon Funk

The Universal Covering of an Inverse Semigroup

We examine an inverse semigroup T in terms of the universal locally constant covering of its classifying topos . In particular, we prove that the fundamental group of coincides with the maximum group image of T. We explain the connection between E-unitary inverse semigroups and locally decidable toposes, characterize E-unitary inverse semigroups in terms of a kind of geometric morphism called a spread, characterize F-inverse semigroups, and interpret McAlister’s “P-theorem” in terms of the universal covering.

Spreads and the symmetric topos II

Abstract We introduce a new intrinsic notion of spread for toposes and geometric morphisms, and use it to give a “topological” characterization of Lawvere distributions on a topos. In the process, we relate spreads to zero-dimensional locales, and establish two new pure/spread factorizations for geometric morphisms. Our results are then applied to the study of the symmetric topos as a generalized lower power locale. In particular, we show that the symmetric topos is part of a Kock-Zoberlein 2-monad on toposes, give a new construction of bicomma squares in which the “lower” leg is an essential geometric morphism, characterize local connectedness in terms of the symmetric topos, and relate the symmetric and bagdomain constructions via the classifier of probability distributions.

Constructive theory of the lower power locale

This paper considers two main aspects of the lower power locale PL(X): first, its relation to the symmetric topos construction of Bunge and Carboni; and second, its points, which, it is shown, are equivalent to the weakly closed sublocales of X with open domain. This is done as part of a more general discussion of arbitrary weakly closed sublocales, including a new characterization using suplattice homomorphisms from 0{X) to Sub{\), and a new proof of a theorem of Jibladze relating them to fi-nuclei.

On a bicomma object condition for KZ-doctrines

Abstract We study Kock–Zoberlein doctrines that satisfy a certain bicomma object condition. Such KZ-doctrines we call admissible. Our investigation is mainly motivated by the example of the symmetric monad on toposes. For an admissible KZ-doctrine, we characterize its algebras in terms of cocompleteness, and we describe its Kleisi 2-category by means of its bifibrations. We obtain in terms of bifibrations a “comprehensive” factorization of 1-cells (and 2-cells). Then we investigate admissibility when the KZ-doctrine is stable under change of base, thus obtaining a characterization of the algebras as linear objects, and the classification of discrete fibrations. Known facts about the symmetric monad are revisited, such as the Waelbroeck theorems. We obtain new results for complete spreads in topos theory. Finally, we apply the theory to the similar examples of the lower power locale and the bagdomain constructions. There is in domain theory an example of a different kind.

The locally connected coclosure of a Grothendieck topos

Abstract Locally connected Grothendieck toposes are shown to be coreflective in Grothendieck toposes. We refer to this coreflection as locally connected coclosure. The locally connected coclosure of a localic topos is localic. A topos and its locally connected coclosure are seen to have equivalent categories of Lawvere distributions. A class of locales is produced whose locally connected coclosures are trivial. We show how the locally connected coclosure can be described in terms of the locally connected coclosure of a localic cover.