## Unique factorisation, refinement properties, and notions of extensivity¶

This project follows on from my paper:

In this project, the aim is to study variants of the well-known categorical property called *extensivity* (see this link). This property captures an essential aspect of categories like **Set, Top, Grph, Pos**, and many others, including dual categories such as **CRing** (commutative rings with unity) or **BDLat** (bounded distributive lattices).

In universal algebra, a property identified by C. Chang, B. Jónsson, and A. Tarski called the *strict refinement property* has significant implications, including a strong form of unique factorisation under direct product. This property can be reformulated as a variant of the classical extensivity condition, allowing it to be handled categorically. Consequently, many key results concerning the strict refinement property can be addressed in a categorical framework.

## Centrality, abelianess, and the commutativity of finite products with coequalisers¶

This project builds upon the following papers:

*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.

These papers explore the categorical algebraic aspects of the commutativity of finite products and coequalisers. Specifically, the property of a category (with finite products and coequalisers) that 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 pointwise product

$$ C_1 \times C_2 \rightrightarrows X_1 \times X_2 \rightarrow Q_1 \times Q_2 $$

is also a coequaliser diagram. This general property is valid in many familiar categorical contexts and is closely related to the classical topic of centrality (in the sense of S.A. Huq).

## Matrix taxonomy¶

Various properties in *categorical algebra* can be presented as a finite matrix of positive integers, such as the matrix below:

$$\left[\begin{array}{ccc|c} 0 & 1 & 1 \\ 1 & 0 & 1 \end{array}\right]$$

These "matrix properties" were introduced by Z. Janelidze in 2008. The above matrix, for example, encodes the property of a category known as a Mal'tsev category. Other properties, such as being unital, strongly unital, subtractive, majority, or arithmetical, also give examples of matrix properties. If *M* is a matrix, the class of all (finitely complete) categories with the matrix property corresponding to *M* is denoted by **mclex**{*M*}. Given two matrices *M* and *N*, we aim to determine whether **mclex**{*M*} is contained in **mclex**{*N*}. In a recent paper with my collaborators, we developed an algorithm to determine such inclusions. Using this algorithm, we can identify all distinct matrix properties up to certain dimensions. The picture below, generated by a computer implementation of our algorithm, shows all distinct matrix properties represented by matrices with at most three rows, with entries either 0 or 1.

Here are links to some published research: