# On the naturalness of Mal'tsev categories

## TL;DR

This paper explores the concept of Mal'tsev categories, highlighting their various characterizations, their role in algebraic structures, and their importance in developing theories of centrality, extensions, and homological lemmas.

## Contribution

It provides a comprehensive analysis of Mal'tsev categories, connecting their multiple characterizations and demonstrating their significance in categorical algebra and homological theories.

## Key findings

- Mal'tsev categories are characterized by various equivalent conditions.
- They serve as the natural setting for centrality and homological concepts.
- The paper clarifies the role of Mal'tsev operations in algebraic theories.

## Abstract

Mal'tsev categories turned out to be a central concept in categorical algebra. On one hand, the simplicity and the beauty of the notion is revealed through a lot of characterizations of different flavour. Depending on the context, one can define Mal'tsev categories as those for which `any reflexive relation is an equivalence'; `any relation is difunctional'; `the composition of equivalence relations on a same object is commutative'; `each fibre of the fibration of points is unital' or `the forgetful functor from internal groupoids to reflexive graphs is saturated on subobjects'. For a variety of universal algebras, these are also equivalent to the existence in its algebraic theory of a Mal'tsev operation, i.e. a ternary operation $p(x,y,z)$ satisfying the axioms $p(x,x,y)=y$ and $p(x,y,y)=x$. On the other hand, Mal'tsev categories have been shown to be the right context in which to develop the theory of centrality of equivalence relations, Baer sums of extensions, and some homological lemmas such as the denormalized $3 \times 3$ Lemma, whose validity in a regular category is equivalent to the weaker `Goursat property', which has also turned out to be of wide interest.