arXiv subjects

Vasileios Aravantinos-Sotiropoulos

Publications and source records attributed to Vasileios Aravantinos-Sotiropoulos.

6 recordsLinked to original sources

arXiv2026

On categories of monads and comonads in double categories

As is well known in the literature, the category Mon(V) of monoids in a monoidal category V satisfies various fundamental categorical properties, at least when the monoidal base V is correspondingly well-behaved. In particular, Mon(V) is monadic over V as soon as free monoids exist, while if V is cocomplete or locally presentable and its tensor b is sufficiently compatible with the appropriate colimits, then Mon(V) inherits the analogous property. In the present work, we extend such results to the context of double categories. More precisely, we identify conditions on a double category D under which one can show that the category Mnd(D) of monads in D is monadic over the category of endomorphisms End(D), is cocomplete or even locally presentable. We also tackle the issue of local presentability in the dual case Cmd(D) of comonads. In these results, our assumptions on the double category D revolve around notions of colimit, in particular those of parallel and stable local colimits, as well as a notion of local presentability of a double category which has been introduced in previous work.

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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.

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arXiv2025

Poset-enriched categories and free exact completions

We give an elementary construction of the exact completion of a weakly lex category for categories enriched in the cartesian closed category $\mathsf{Pos}$ of partially ordered sets. Paralleling the ordinary case, we characterize categories which arise as such completions in terms of projective objects. We then apply the results to categories of Eilenberg-Moore algebras for monads on $\mathsf{Pos}$. In particular, we show that every variety of ordered algebras is the exact completion of a subcategory on certain free algebras, thereby answering a question of A. Kurz and J. Velebil.

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arXiv2025

Sweedler theory for double categories

In this work, we establish certain enrichments of dual algebraic structures in the setting of monoidal double categories. In more detail, we obtain a tensored and cotensored enrichment of monads in comonads, as well as a tensored and cotensored enrichment of modules in comodules, under very general conditions on the surrounding double category. These include `monoidal closedness' and `local presentability', leading classical notions which are here introduced in the double categorical context. Furthermore, we show that in this setting, the actual fibration of modules over monads is itself enriched in the opfibration of comodules over comonads. Applying this abstract double categorical framework to the setting of V-matrices, one directly obtains a many-object generalization of the known enrichment of modules over monoids in comodules over comonoids in a monoidal category V, which was originally induced by Sweedler's measuring k-coalgebras in vector spaces. In the present setting the result involves V-enriched modules of V-categories and V-enriched comodules of V-cocategories.

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arXiv2021

The Exact Completion for Regular Categories enriched in Posets

We construct an exact completion for regular categories enriched in the cartesian closed category $\mathsf{Pos}$ of partially ordered sets and monotone functions by employing a suitable calculus of relations. We then characterize the embedding of any regular category into its completion and use this to obtain examples of concrete categories which arise as such completions. In particular, we prove that the exact completion in this enriched sense of both the categories of Stone and Priestley spaces is the category of compact ordered spaces of L. Nachbin. Finally, we consider the relationship between the enriched exact completion and categories of internal posets in ordinary categories.

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arXiv2019

Projective Covers of 2-star-permutable Categories

We introduce the notion of star-symmetry for relations in a multi-pointed category and use it to obtain a characterization of the projective covers of 2-star-permutable categories. This generalizes the results of Rosický-Vitale for regular Mal'tsev categories, as well as those of Gran-Rodelo for regular subtractive categories. We apply the characterization in terms of star-symmetry to recover the syntactic conditions defining E-subtractive varieties in the sense of Ursini.

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