Matrix taxonomy and Bourn localization

TL;DR

This paper extends an algorithm to analyze matrix properties of categories, including pointedness and Bourn localizations, enabling new characterizations of certain categorical classes through computational methods.

Contribution

It introduces an extended algorithm for matrix property implications, including pointedness and Bourn localizations, with a computer implementation for visualization and analysis.

Findings

Characterization of Bourn localizations via matrix properties

Identification of implications among matrix properties

Computational grouping of properties by localizations

Abstract

In a recent paper, an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices -- the so called `matrix properties'. In this paper we extend this algorithm to include matrix properties involving pointedness of a category, such as the properties of a category to be unital, strongly unital or subtractive, for example. Moreover, this extended algorithm can also be used to determine whether a given matrix property is the Bourn localization of another, thus leading to new characterizations of Mal'tsev, majority and arithmetical categories. Using a computer implementation of our algorithm, we can display all such properties given by matrices of fixed dimensions, grouped according to their Bourn localizations, as well as the implications between them.