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

This project follows on from my paper:

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

In the subject of universal algebra, there is a property which has been identified and studied first by C. Chang, B. Jónsson, A. Tarski called the *strict refinement property* - see this paper. This property has several important consequences, one of which is a strong form of unique factorisation under direct product. It turns out that this property can be reformulated as a variant of the classical extensivity condition, and then can be dealt with purely categorically. Moreover, some of the main results surrounding the strict refinement property can then be effectively dealt with in a categorical manner.

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

This project builds upon the papers given below.

These papers study the categorical algebraic aspects of the commutativity of finite products and coequalisers. That is, 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 a coequaliser diagram. Apart from being a very general property in categorical algebra, it turns out that commutativity of finite products with coequlaisers is closely related to topics involving abelianization and centrality (such as in the category of groups, for example).

## Matrix taxonomy¶

This interdisciplinary project has lead to three papers (together with my collaborators) which have been submitted for publication, and there is currently still research underway in exploring its various aspects. Various properties in *categorical algebra* may be presented as a finite matrix of positive integers such at the matrix below:

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

This matrix captures the essential categorical property of a category to be what is know as a Mal'tsev category in categorical algebra. If *M* is a matrix, then the class of all (finitely complete) categories which have the matrix property corresponding to *M* is denoted by **mclex**{*M*}. Given two such matrices *M* and *N*, we are interested in determining whether **mclex**{*M*} is contained in **mclex**{*N*}. In a recent paper together with my collaborators, we have determined a simple algorithm for determining such inclusions. Using this algorithm we can determine all distinct matrix properties up to some given dimensions. For instance, the picture below was generated by a computer implementation of our algorithm, which indicates all distinct matrix properties represented by matrices with at at most three rows, whose entries are either 0 or 1.

Here are limks to some published research.