Re: Products of epimorphisms

[Clemens.BERGER@unice.fr]

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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