UCLouvain-ULB-VUB Category Theory Seminar

by Andrea Montoli (University of Milan), Tim Van der Linden (Université catholique de Louvain)

Monday 23 Mar 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. Andrea Montoli (University of Milan) : On the categorical behaviour of V-groups

Abstract :We will consider compatible group structures on a V-category, where V is a quantale, and we will explore some categorical properties of such groups. Examples of such structures are preordered groups, metric and ultrametric groups, probabilistic (ultra)metric groups. In particular, we show that, when V is a cartesian quantale, symmetric V-groups satisfy very strong categorical-algebraic properties, typical of the category of groups, while the whole category of V-groups satisfies similar properties relatively to a suitable class of split epimorphisms, similarly to what happens for the category of monoids.

Joint work with Maria Manuel Clementino.

5.25 p.m. Tim Van der Linden (UCLouvain and VUB): The bilinear product: an intrinsic approach to tensor products

Abstract :In the context of a Janelidze-Márki-Tholen semi-abelian category, we establish an intrinsic definition of a bilinear product, a tensor-like operation on objects of a category, constructed in terms of limits and colimits. Given two objects in the category, their bilinear product is the abelian object obtained as the cosmash product in the category of two-nilpotent objects of the reflections of these objects. In many concrete cases, this operation, applied to a pair of abelian objects, captures a classical tensor product. We explain this by means of a recognition theorem, which states that any symmetric, bi-cocontinuous bifunctor on an abelian variety of algebras can be recovered as the bilinear product within a suitable semi-abelian variety, namely of algebras over a certain two-nilpotent operad. In other words, the extra structure carried by such a bifunctor on the abelian variety (for instance, a tensor product, known in the literature) is encoded by means of a surrounding semi-abelian variety whose abelian core is the original variety.

We illustrate the construction with several examples, develop its basic properties, and compare it to the semi-abelian analogue of the Brown-Loday non-abelian tensor product. As an application, we present a categorical version of Ganea's six-term exact homology sequence.

This is joint work with Bo Shan Deval and Manfred Hartl, based on the preprint arXiv:2512.03951.