Products of epimorphisms

Products of epimorphisms

[David Yetter]

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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Re: Products of epimorphisms

[Peter Johnstone]

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
>
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Products of epimorphisms

[Clemens.BERGER]

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
>
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]