Enriching Vitale’s thesis

by Dr Vasileios Aravantinos Sotiropoulos

Monday Mar 24, 2025, 3:15 PM → 4:15 PM Europe/Brussels

B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

The notions of regular and (Barr-)exact category are ubiquitous in Category Theory, appearing in wide-ranging contexts such as (semi-)abelian categories and elementary toposes. In his 1994 PhD thesis, Enrico Vitale (working jointly with Aurelio Carboni) describes two processes of universally embedding a category with only weak finite limits into a regular or exact one respectively. These regular and exact completions have by now become standard tools and have led to numerous applications.

In this talk, we will begin by looking back at Vitale's thesis and some of the applications that have flowed from it. Subsequently, we will switch our focus to categories enriched over posets, a context which has been the subject of active research in recent years. After recalling the relevant notions of poset-enriched regularity and exactness, we will describe how the constructions of the two associated completions, as described in Vitale's thesis, can be modified to work in this enriched setting. The analogy with the ordinary case is straightforward enough that one might hope to retain analogues of some of the applications as well. We illustrate this last point by discussing a monadicity theorem over the category of posets.