arXiv subjects

Panagis Karazeris

Publications and source records attributed to Panagis Karazeris.

4 recordsLinked to original sources

arXiv2026

Locales in presheaf toposes vs. presheaves of locales

By a well-known characterisation, in a presheaf topos every internal suplattice is a presheaf of suplattices, but not every presheaf of suplattices is an internal suplattice (and similarly for frames). In this paper, we construct the free internal suplattice/frame on an presheaf of suplattices/frames, yielding a left adjoint to the forgetful functor from the respective internal structures to presheaves of structures. The description of this left adjoint has also appeared in recent work of Henry and Townsend, in connection to a different universal property, namely that of turning a lax natural transformation between poset-enriched functors to a strict one. As an application of our construction, we investigate conditions on frames internal to a presheaf topos, such as being locally compact, compact, stably locally compact or Hausdorff, in terms of properties of their sections in the base topos. In the first three cases, it is necessary that all the sections have the respective properties, while the Hausdorff property is not transferred to the sections. Moreover for local compactness it is necessary that the transition maps preserve the way-below relation. Finally, for an internal locally compact frame in presheaves we analyse the connection of its way-below relation to the respective relations of its sections.

math.CT↗

arXiv2024

Filtral pretoposes and compact Hausdorff locales

The category of compact Hausdorff locales is a pretopos which is filtral, meaning that every object is covered by one whose subobject lattice is isomorphic to the lattice of filters of complemented elements. We show that any filtral pretopos satisfying some mild additional conditions can be embedded into the category of compact Hausdorff locales. This result is valid in the internal logic of any topos. Assuming the principle of weak excluded middle and the existence of copowers of the terminal object in the pretopos, the image of the embedding contains all spatial compact Hausdorff locales. The notion of filtrality was introduced by V. Marra and L. Reggio (Theory Appl. Categ., 2020) to characterise the category of compact Hausdorff spaces within the class of pretoposes. Our results can be regarded as a constructive extension of the aforementioned characterisation, avoiding reference to points. If the ambient logic is classical, i.e. it satisfies excluded middle, and the prime ideal theorem for Boolean algebras holds, we obtain as a corollary the characterisation of compact Hausdorff spaces in op. cit.

math.CT↗

arXiv2020

Regular and effective regular categories of locales

We examine the analogues for the respective categories of locales of two well-known results about regularity and effectiveness of some categories of spaces. We show that the category of compact regular locales is effective regular (=Barr-exact). We also show that the category of compactly generated Hausdorff locales is regular, provided that it is coreflective within Hausdorff locales. We do not appeal to the existence of points (which would render the two results trivial) but rely on the treatment of the subject by methods that are valid in the internal logic of a topos. On the course to the result about compactly generated locales we arrive at a generalization of a result of B. Day and R. Street, deriving regularity for a cocomplete category containing a dense regular subcategory closed under finite limits and colimits and satisfying a certain compatibility condition of pullbacks with appropriate colimits.

math.CT↗

arXiv2009

Final coalgebras in accessible categories

We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in this case. On the other hand, there are interesting examples of final coalgebras beyond the realm of l.f.p. categories to which our results apply. We rely on ideas developed by Tom Leinster for the study of self-similar objects in topology.

math.CT↗