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In category theory, you see the idea of a class of epimorphisms being stable under pullback. For example, in a regular category, the class of regular epimorphisms is closed under pullback. Every place I've seen the notion of pullback-stability, it's always a part of a bigger definition, such as regular category, or Grothendieck topology. Is there some bigger significance to the idea? Is there a theory for pullback-stable classes of epimorphisms?

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    I wouldn't call this a theory but these epimorphisms are called "epimorphismes universels" in SGA 4 (I.10.3).2011-02-04

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