Majority categories

TL;DR

This paper introduces majority categories as a categorical framework capturing properties of algebraic and geometric categories, extending concepts like Mal'tsev categories, and characterizes arithmetical categories via majority and Barr-exact conditions.

Contribution

It defines majority categories, explores their properties, and establishes their connection to Mal'tsev and arithmetical categories, providing new insights into categorical algebra.

Findings

Algebraic majority categories include lattices, Boolean, and Heyting algebras.

Many geometric categories are comajority, with duals being majority categories.

A category is arithmetical iff it is Barr-exact, Mal'tsev, and a majority category.

Abstract

We introduce the notion of a majority category --- the categorical counterpart of varieties of universal algebras admitting a majority term. This notion can be thought to capture properties of the category of lattices, in a way that parallels how Mal'tsev categories capture properties of the category of groups. Among algebraic majority categories are the categories of lattices, Boolean algebras, and Heyting algebras. Many geometric categories such as the category of topological spaces, metric spaces, ordered sets, any topos, ect., are comajority categories (i.e.~their duals are majority categories), and we show that, under mild assumptions, the only categories which are both majority and comajority, are the preorders. Mal'tsev majority categories provide an alternative generalization of arithmetical categories to protoarithmetical categories in the sense of Bourn. We show that every Mal'tsev majority category is protoarithmetical, provide a counter-example for the converse implication, and show that in the Barr-exact context, the converse implication also holds. We can then conclude that a category is arithmetical if and only if it is a Barr-exact Mal'tsev majority category, recovering in the varietal context a well-known result of Pixley.