Dear David, your property holds in any regular category provided f and g are regular epis, and so in any elementary topos provided f and g are mere epis. Indeed, your square is a pullback square of regular epis, and in a regular category, any such is also a pushout square. A slightly more general context where your property holds is a finitely complete category with a strong epi-mono factorisation system for which strong epis are closed under pullback along monos and closed under cartesian product. All the best, Clemens. Le 2018-01-19 20:20, David Yetter a ??crit??:
Dear fellow category theorists,
I'm interested in finding out at what level of generality a result that plainly holds in Sets (and Sets^op) is true:
Given two epimorphisms f:A-->>B and g:C-->D, the square formed by f x 1_C, 1_A x g, f x 1_D and 1_B x D is a pushout.
I'd like it to be true (at least) in toposes and I think I have an element-wise proof (but don't remember the details of the semantics given by, for instance, Osius, well enough to be sure I've really proven the result in all toposes -- it's been years since I thought seriously about that sort of thing).
And, is there anywhere in the literature that this occurs? It feels like the sort of thing that would have been known long ago.
Best Thoughts, David Yetter Professor of Mathematics Kansas State University
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