# Some stability properties of epimorphism classes

## Dali Zangurashvili

It is proved that in any pointed category with pullbacks, coequalizers and regular epi-mono factorizations, the class of regular epimorphisms is stable under pullback along the so-called balanced effective descent morphisms. Here balanced'' can be omitted if the category is additive. A balanced effective descent morphism is defined as an effective descent morphism $p:E\rightarrow B$ such that any subobject of $E$ is a pullback of some morphism along $p$. It is shown that, in any category with pullbacks and coequalizers, the class of effective descent morphisms is stable under pushout if and only if any regular epimorphism is an effective descent morphism. Moreover, it is shown that the class of descent morphisms is stable under pushout if and only if the class of regular epimorphisms is stable under pullback.

Keywords: (effective) descent morphism, balanced morphism, factorization system, stability under pullback/pushout

2010 MSC: 18A20, 18A32, 18A30, 18C20

Theory and Applications of Categories, Vol. 29, 2014, No. 1, pp 1-16.

Published 2014-01-19.

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