The matrix taxonomy of finitely complete categories

TL;DR

This paper develops a taxonomy of finitely complete categories using matrix properties, providing an algorithm to determine implications among these properties and exploring the structure of their classes.

Contribution

It introduces an algorithmic approach to classify finitely complete categories via matrix properties and analyzes the poset structure of these classes.

Findings

Algorithm for implication decision among matrix properties

Structure analysis of the poset of matrix classes

Identification of key classes like Mal'tsev and majority categories

Abstract

This paper is concerned with the taxonomy of finitely complete categories, based on 'matrix properties' - these are a particular type of exactness properties that can be represented by integer matrices. In particular, the main result of the paper gives an algorithm for deciding whether a conjunction of such properties implies another such property. Computer implementation of this algorithm allows one to peer into the complex structure of the poset of `matrix classes', i.e., the poset of all collections of finitely complete categories determined by matrix properties. Among elements of this poset are the collections of Mal'tsev categories, majority categories, (finitely complete) arithmetical categories, as well as finitely complete extensions of various classes of varieties defined by a special type of Mal'tsev conditions found in the literature.