Research Projects¶
My research currently develops several topics at the intersection of category theory and algebra, some specific projects I am working on are listed below.
Unique Factorisation, Refinement Properties, and Notions of Extensivity¶
The aim is to study categorical variants of the classical property known as extensivity (see nLab entry). Extensivity captures essential structural aspects of categories such as Set, Top, Grph, Pos, and duals of certain algebraic categories like like CRing (commutative rings with unity) and BDLat (bounded distributive lattices).
In universal algebra, the strict refinement property—identified by Chang, Jónsson, and Tarski—implies a strong form of unique factorisation with respect to direct products. Through a notion of extensive morphism (see article below), this property can be formulated categorically as a variant of coextensivity. As a result, many theorems involving refinement and factorisation can be approached categorically and generalised.
Papers in the project¶
- M-coextensive objects and the strict refinement property, Journal of Pure and Applied Algebra, 224(10), 2020.
- (with E. Theart) On extensivity and coextensivity of morphisms, Theory and Applications of Categories, Vol. 44, 2025, No. 21, pp 617–642.
Commutativity of Finite Products with Qoutients¶
These works explore the property, which is common in algebra, that products commute with quotients. Loosely speaking, this is a categorical formulation:
$$ (X_1 / C_1) \times (X_2 / C_2) \cong (X_1 \times X_2) / (C_1 \times C_2) $$
For groups, it is well known that if \(N_1\) is a normal subgroup of \(G_1\) and \(N_2\) is a normal subgroup of \(G_2\), then there is a canonical isomorphism:
$$ (G_1 / N_1) \times (G_2 / N_2) \cong (G_1 \times G_2) / (N_1 \times N_2) $$
A general categorical notion of qoutient is coequaliser, so that we may ask the question: for which categories \(\mathbb{C}\) do we have that when given two coequaliser diagrams
$$ C_1 \rightrightarrows X_1 \rightarrow Q_1 \quad \text{and} \quad C_2 \rightrightarrows X_2 \rightarrow Q_2 $$
their product
$$ C_1 \times C_2 \rightrightarrows X_1 \times X_2 \rightarrow Q_1 \times Q_2 $$
that is again coequaliser. This property turns out to be very common (it is implied throughout a wide variety of categorical-algebraic contexts), and closely related to the classical topic of centrality of morphisms.
Papers in the project¶
- Centrality and the commutativity of finite products with coequalisers, Theory and Applications of Categories, 39(13), 423–443, 2023.
- Products and coequalizers in pointed categories, Theory and Applications of Categories, 34, 1386–1400, 2019.
Matrix Taxonomy in Categorical Algebra¶
Many properties in categorical algebra can be encoded as finite matrices of natural numbers. In this paper, my collaborators and I developed an algorithm to determine these inclusion relationships. This allows us to classify all distinct matrix properties up to a given size.
The diagram below, generated by our implementation, visualises all distinct matrix properties defined by matrices with up to three rows:
Papers in this project¶
- (with P.-A. Jacqmin, Z. Janelidze) The matrix taxonomy of finitely complete categories, Theory and Applications of Categories, Vol. 38, 737–790, 2022.
- (with P.-A. Jacqmin) Matrix taxonomy and Bourn localization, Applied Categorical Structures, 2022.
- (with P.-A. Jacqmin, Z. Janelidze, E. van der Walt) On binary matrix properties, Quaestiones Mathematicae, 2023.
- (with P.-A. Jacqmin) Partial Algebras and Implications of (Weak) Matrix Properties, Applied Categorical Structures, 32, 34, 2024.
- (with P.-A. Jacqmin) When a matrix condition implies the Mal’tsev property, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume LXVI (2025), No. 3, pp. 32–64.
Additional material: