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Weakly Mal'tsev categories and strong relations

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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.

http://www.tac.mta.ca/tac/volumes/27/5/27-05.dvi

http://www.tac.mta.ca/tac/volumes/27/5/27-05.ps

http://www.tac.mta.ca/tac/volumes/27/5/27-05.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/5/27-05.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/5/27-05.ps

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