# Characterizations of majority categories

## TL;DR

This paper extends algebraic characterizations of majority terms to regular categories, establishing categorical versions of key theorems like Bergman's Double-projection and Pixley's congruence equation.

## Contribution

It provides categorical analogues of algebraic majority characterizations, including a version of Bergman's theorem and the Chinese Remainder Theorem, for regular categories.

## Key findings

- A regular category is a majority category iff subobjects are determined by two-fold projections.
- Established a categorical version of Bergman's Double-projection Theorem.
- Characterized regular majority categories via Pixley's congruence equation.

## Abstract

In universal algebra, it is well known that varieties admitting a majority term admit several Mal'tsev-type characterizations. The main aim of this paper is to establish categorical counterparts of some of these characterizations for regular categories. We prove a categorical version of Bergman's Double-projection Theorem: a regular category is a majority category if and only if every subobject $S$ of a finite product $A_1 \times A_2 \times \cdots \times A_n$ is uniquely determined by its two-fold projections. We also establish a categorical counterpart of the Pairwise Chinese Remainder Theorem for algebras, and characterize regular majority categories by the classical congruence equation $\alpha \cap (\beta \circ \gamma) = (\alpha \cap \beta) \circ (\alpha \cap \gamma)$ due to A.F.~Pixley.