Re: Products of epimorphisms

[Peter Johnstone <ptj@dpmms.cam.ac.uk>]

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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