* Re: Products of epimorphisms
2018-01-19 19:20 Products of epimorphisms David Yetter
@ 2018-01-20 10:57 ` Peter Johnstone
2018-01-20 13:49 ` Clemens.BERGER
1 sibling, 0 replies; 3+ messages in thread
From: Peter Johnstone @ 2018-01-20 10:57 UTC (permalink / raw)
To: David Yetter; +Cc: categories
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/ ]
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: Products of epimorphisms
2018-01-19 19:20 Products of epimorphisms David Yetter
2018-01-20 10:57 ` Peter Johnstone
@ 2018-01-20 13:49 ` Clemens.BERGER
1 sibling, 0 replies; 3+ messages in thread
From: Clemens.BERGER @ 2018-01-20 13:49 UTC (permalink / raw)
To: David Yetter; +Cc: categories
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/ ]
^ permalink raw reply [flat|nested] 3+ messages in thread