UCLouvain-ULB-VUB Category Theory Seminar

by Prof. Diana Rodelo (Universidade do Algarve), George Peschke (University of Alberta), Prof. Simon Henry (University of Ottawa)

Wednesday 3 Jun 2026, 16:15 → 18:15 Europe/Brussels

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

4.15 p.m. : Prof. Diana Rodelo (Universidade do Algarve) « Some remarks on (weakly) protomodular objects in unital categories ».

Abstract: We recall a purely categorical characterisation of groups amongst monoids which was obtained by considering a local (or object-wise) version of protomodularity. This led to the notion of protomodular object in a (not necessarily protomodular) category. Groups are precisely the protomodular objects in Mon, the category of monoids [1]. It was later shown in [2] that groups are also the weakly protomodular objects (a weaker notion than that of protomodular object) in Mon. The coincidence of weakly protomodular and protomodular objects in Mon is not a unique case. Indeed, there are other unital categories, besides Mon, where this coincidence holds. In this talk we compare the concepts of protomodular and weakly protomodular objects within the context of unital categories and show that these two notions are generally distinct. This talk is based on [3].

[1] A. Montoli, D. Rodelo, and T. Van der Linden, Two characterisations of groups amongst monoids, J. Pure Appl. Algebra 222 (2018), 747-777.

[2] X. García-Martinez, A new characterisation of groups amongst monoids, Appl. Categ. Structures 25 (2017), no. 4, 659--661.

[3] X. García-Martinez, A. Montoli, D. Rodelo, T. Van der Linden, A comparison between weakly protomodular and protomodular objects in unital categories, accepted for publication in Communications in Algebra.

 

4.55 p.m. Prof. George Peschke (University of Alberta): "Galois adjunctions »

Abstract: We introduce the concept of ‘Galois adjunction’ as a categorification of ‘Galois connection’ between posets. Examples of Galois adjunctions include (co-)reflectors, and the limit/colimit adjunction of a small diagram in a bicomplete category. Along with every Galois adjunction comes a category of ‘Galois adjunction pairs’. Special cases include ‘bicartesian squares’ and ‘short exact sequences’.

 

5.35 p.m. Prof. Simon Henry (University of Ottawa): "Rewriting for Quasicategories »
 

Abstract: Rewriting is a set of tools that allows given a "nice enough" presentation of an algebraic structure - for us a monoid or a category - allows to extract homotopy theoretic information from the monoid, for example computing its homology. In this talk I'll give a quick introduction to rewriting and explain how from these methods we can actually get an explicit "homotopy coherent" presentation of the monoid or the category as an infinity category. This allows to give easy proofs of a lot of manipulations in higher category theory. Surprisingly the proof of the main result appeared in a 35 years old paper of Brown - predating the development of the modern theory of infinity categories, and because of that the result itself went apparently unnoticed.