UCLouvain-ULB-VUB Category Theory Seminar

by Sacha Ikonicoff (Université de Strasbourg), Tim Van der Linden (Université catholique de Louvain)

Monday 19 Jan 2026, 16:15 → 18:15 Europe/Brussels

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

16:15 Sacha Ikonicoff (Université de Strasbourg) : Tangent structures and operads

Abstract: Tangent categories provide an axiomatisation for the notion of tangent bundle, and provide tools of a certain geometric flavour to study objects of a category. The aim of this talk is to describe certain tangent structures which appear on the categories of algebras over an operad (and on the opposite category). An operad is an algebraic device that encode types of (multilinear) operations satisfying (multilinear) relation. As such, each operad has an associated category of algebras, which forms a variety of universal algebra in a certain sense. For a given operad P, we will study a tangent structure on the category of P-algebras given by a notion of semidirect product, sometimes called the "algebraic" tangent structure, as well as an adjoint tangent structure on the opposite category, given by a generalisation of Kähler differentials, sometimes referred to as the "geometric" tangent structure for "P-affine schemes", by analogy with the algebraic geometry viewpoint. This gives a common framework for studying aspects of algebraic geometry, non-commutative geometry, and certain notions of geometry for Lie algebras, Poisson algebras, and so on.
This talk will start with a leisurely introduction to the notion of operads. We will then introduce tangent categories, reviewing both the archetypal example of smooth manifolds and the more algebraic example of commutative algebras. Then we will see how the latter example can be generalised to algebras over any operad.

This is the result of work in collaboration with Marcello Lanfranchi and JS Lemay which be found on arXiv under reference 2303.05434 and 2108.04304.

17:30 Tim Van der Linden (UCLouvain) : 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 (which will be updated soon).