UCLouvain-ULB-VUB Category Theory Seminar

by Dr Nathanael Arkor (Talinn University of Technology), Prof. Vincenzo Marra (University of Milan)

Monday Jun 2, 2025, 4:15 PM → 6:15 PM Europe/Brussels

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

Vincenzo Marra: An extension of Stone-Gelfand Duality
 

Abstract: In a recent paper co-authored with M. Abbadini and L. Spada [1], we extended Stone-Gelfand duality to compact Hausdorff spaces equipped with additional arithmetic structure. That structure is determined on a space by embedding it into a Tychonoff cube. Dually, one can think of this generalisation as weakening the structure of real linear spaces to that of Abelian groups. In this talk I plan to describe this duality result, emphasising an intriguing connection with lax comma 2-categories which is as yet ill understood. 
[1] M. Abbadini, V. Marra, and L. Spada, Stone-Gelfand duality for metrically complete lattice-ordered groups, Adv. Math. 461, 2025, 33 pp.

 

Nathanael Arkor: Hierarchical algebra and enriched categories
 

Abstract: In classical universal algebra, one studies structures equipped with operations whose arities are taken from a fixed set of sorts. However, there are many examples where the sorts do not merely form a set, but rather are equipped with further algebraic structure. Following Lawvere's example, we are led to investigate such structures from the perspective of categorical algebra. We shall see that this hierarchical algebraic structure may be viewed fruitfully from the perspective of enriched category theory. Unusually, however, the appropriate base of enrichment is not a monoidal category, but rather a skew-monoidal category. If time permits, I will explain how variable-binding operations may also be captured in this setting, leading to a two-dimensional perspective on parametric polymorphism.