An embedding theorem for regular Mal'tsev categories

TL;DR

This paper extends Lubkin's embedding theorem to regular Mal'tsev categories, providing a faithful embedding into a power of a locally finitely presentable category that preserves key structures.

Contribution

It introduces a non-abelian analogue of Lubkin's theorem, embedding small regular Mal'tsev categories into powers of a specific regular Mal'tsev category.

Findings

Embedding preserves finite limits, isomorphisms, and regular epimorphisms.

The number of subobjects of the terminal object determines the embedding power.

Provides a non-abelian generalization of Barr's embedding theorem.

Abstract

In this paper, we obtain a non-abelian analogue of Lubkin's embedding theorem for abelian categories. Our theorem faithfully embeds any small regular Mal'tsev category $\mathbb{C}$ in an $n$-th power of a particular locally finitely presentable regular Mal'tsev category. The embedding preserves and reflects finite limits, isomorphisms and regular epimorphisms, as in the case of Barr's embedding theorem for regular categories. Furthermore, we show that we can take $n$ to be the (cardinal) number of subobjects of the terminal object in $\mathbb{C}$.