# $\mathcal{M}$-coextensivity and the strict refinement property

## TL;DR

This paper introduces the concept of $\\mathcal{M}$-coextensivity in categories to unify and analyze the strict refinement property in universal algebra, connecting it with categorical notions of extensivity and algebraic decomposability.

## Contribution

It defines $\mathcal{M}$-coextensive objects in arbitrary categories and relates them to the strict refinement property and algebraic decomposability, providing a categorical framework for these concepts.

## Key findings

- $\\mathcal{M}$-coextensive objects correspond to algebras with the strict refinement property.
- In varieties of algebras, projection-coextensive objects are exactly those with the strict refinement property.
- Objects with global support in certain categories have the strict refinement property.

## Abstract

The notion of an $\mathcal{M}$-coextensive object is introduced in an arbitrary category $\mathbb{C}$, where $\mathcal{M}$ is a distinguished class of morphisms from $\mathbb{C}$. This notion allows for a categorical treatment of the strict refinement property in universal algebra, and highlights its connection with extensivity in the sense of Carboni, Lack and Walters. If $\mathcal{M}$ is the class of all product projections in a variety of algebras $\mathbb{C}$, then the $\mathcal{M}$-coextensive (or projection-coextensive) objects in $\mathbb{C}$ turn out to be precisely those algebras which have the strict refinement property. If $\mathcal{M}$ is the class of surjective homomorphisms in the variety, then the $\mathcal{M}$-coextensive objects are precisely those algebras which have directly-decomposable (or factorable) congruences. In exact Mal'tsev categories, every centerless object with global support has the strict refinement property. We will also show that in every exact majority category, every object with global support has the strict refinement property.