Mal'tsev objects, $R_1$-spaces and ultrametric spaces

TL;DR

This paper introduces Mal'tsev objects and their duals in categories, characterizes Mal'tsev categories, and identifies co-Mal'tsev objects as $R_1$-spaces and ultrametric spaces in topological and metric categories.

Contribution

It defines Mal'tsev and co-Mal'tsev objects in categories and characterizes their instances as $R_1$-spaces and ultrametric spaces, linking categorical properties to topological and metric structures.

Findings

Mal'tsev objects form a coreflective subcategory in well-powered regular categories.

Co-Mal'tsev objects in topological spaces are exactly $R_1$-spaces.

Co-Mal'tsev objects in metric spaces are precisely ultrametric spaces.

Abstract

In this paper we introduce a notion of Mal'tsev object, and the dual notion of co-Mal'tsev object, in a general category. In particular, a category $\mathbb{C}$ is a Mal'tsev category if and only if every object in $\mathbb{C}$ is a Mal'tsev object. We show that for a well-powered regular category $\mathbb{C}$ which admits coproducts, the full subcategory of Mal'tsev objects is coreflective in $\mathbb{C}$. We show that the co-Mal'tsev objects in the category of topological spaces and continuous maps are precisely the $R_1$-spaces, and that the co-Mal'tsev objects in the category of metric spaces and short maps are precisely the ultrametric spaces.