Dear David, Yes, this is true for regular epis in any cartesian closed category. I don't know a published proof, but here is the argument: if h: A x C --> E factors through 1_A x g, then its transpose C --> E^A factors through g, and so coequalizes any pair R ==> C of which g is a coequalizer. But if h also factors through f x 1_C, then its transpose factors through E^B --> E^A, which is monic (since E^(-), being self-adjoint on the right, sends epis to monos), so the induced C --> E^B also coequalizes R ==> C and hence factors through g. Hence h factors through f x g. In fact, as is clear from the above argument, we need only one of f and g to be regular epic. Best regards, Peter Johnstone On Fri, 19 Jan 2018, David Yetter wrote:
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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