# Weakly Mal'tsev categories and strong relations

## Zurab Janelidze and Nelson Martins-Ferreira

We define a strong relation in a category $\mathbb{C}$ to be a span which is orthogonal'' to the class of jointly epimorphic pairs of morphisms. Under the presence of finite limits, a strong relation is simply a strong monomorphism $R\rightarrow X\times Y$. We show that a category $\mathbb{C}$ with pullbacks and equalizers is a weakly Mal'tsev category if and only if every reflexive strong relation in $\mathbb{C}$ is an equivalence relation. In fact, we obtain a more general result which includes, as its another particular instance, a similar well-known characterization of Mal'tsev categories.

Keywords: weakly Mal'tsev category, Mal'tsev category, difunctional relation, factorization system

2010 MSC: 18C99, 18A20

Theory and Applications of Categories, Vol. 27, 2012, No. 5, pp 65-79.

Published 2012-07-27.

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