# The Lawvere condition and a classification theorem for Mal'tsev categories

## TL;DR

This paper establishes a classification scheme for Mal'tsev categories based on classifying classes of spans, using the Lawvere condition to characterize different types of Mal'tsev-like categories through internal categorical structures.

## Contribution

It introduces a new classification theorem for Mal'tsev, naturally Mal'tsev, and weakly Mal'tsev categories using classifying classes of spans and a compatibility condition, unifying main characterizations.

## Key findings

- Class of all spans characterizes naturally Mal'tsev categories.
- Relations characterize Mal'tsev categories.
- Strong relations characterize weakly Mal'tsev categories.

## Abstract

A classification theorem for three different sorts of Mal'tsev categories is proven. The theorem provides a classification for Mal'tsev category, naturally Malt'sev category, and weakly Mal'tsev category in terms of classifying classes of spans. The class of all spans characterizes naturally Mal'tsev categories. The class of relations (i.e. jointly monomorphic spans) characterizes Mal'tsev categories. The class of strong relations (i.e. jointly strongly monomorphic spans) characterizes weakly Mal'tsev categories. The result is based on the uniqueness of internal categorical structures such as internal category and internal groupoid (Lawvere condition). The uniqueness of these structures is viewed as a property on their underlying reflexive graphs, restricted to the classifying spans. The class of classifying spans is combined, via a new compatibility condition, with split squares. This is analogous to orthogonality between spans and cospans. The result is a general classifying scheme which covers the main characterizations for Mal'tsev like categories. The class of positive relations has recently been shown to characterize Goursat categories and hence it is a new example that fits in this general scheme.