From: Jaap van Oosten <J.vanOosten@uu.nl>
To: Reinhard Boerger <Reinhard.Boerger@FernUni-Hagen.de>, categories@mta.ca
Subject: Re: Famous unsolved problems in ordinary category theory
Date: Thu, 11 Jun 2009 13:30:55 +0200 [thread overview]
Message-ID: <E1MEk5t-0000Hr-1z@mailserv.mta.ca> (raw)
Reinhard Boerger wrote:
> Dear categorists,
>
> When I read the question for the first time, I did not know such a problem.
> Moreover, my impression was that in category theory one often finds new
> results, which had not been conjectured before. Sometimes an important part
> of the work is even to develop the right notions. This may explain that
> there are less important well-known problems in category theory than in
> other areas.
>
> Nevertheless, I remember a problem that can be easily formulated in pure
> category and is still unsolved as far as I know. Bur it does not seem vastly
> distributed. Cantor's diagonal says that says that the power set always is
> of larger cardinality as the original set. Gavin Wraith suggested the
> following generalization to topoi: If for two objects A,B there is a
> monomorphism A^B>->B, is there also a monomorphism A>->1? This looks like a
> meaningful analogue, and I have not seen an answer in the meantime. The
> question can even be asked not only in a topos, but in every cartesian
> closed category. Does anybody know anything about progress?
>
Dear Professor Boerger,
there are counterexamples to this in elementary topoi. In the effective
topos there are nontrivial objects X such that 2^X is isomorphic to 2
(for example, one can take the object R of real numbers for X; this
gives a mono 2^R>->R), and the inclusion N-->N^{P(N)} is an isomorphism
(giving a mono N^{P(N)}>->P(N) , where P(N) is the power object of N).
I believe the example of 2^R>->R also holds in sheaf toposes where
(internally) the object of functions R^R coincides with the object of
continuous functions (since R is always connected).
Best, Jaap van Oosten
> Greetings
> Reinhard
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next reply other threads:[~2009-06-11 11:30 UTC|newest]
Thread overview: 18+ messages / expand[flat|nested] mbox.gz Atom feed top
2009-06-11 11:30 Jaap van Oosten [this message]
-- strict thread matches above, loose matches on Subject: below --
2009-06-09 15:47 Steve Vickers
2009-06-09 13:35 Reinhard Boerger
2009-06-06 9:18 soloviev
2009-06-06 3:59 Bhupinder Singh Anand
2009-06-06 1:35 Hasse Riemann
2009-06-05 22:36 Robin Cockett
2009-06-05 14:17 Thomas Streicher
2009-06-05 11:07 Ronnie Brown
2009-06-05 8:41 Paul Taylor
2009-06-05 4:10 John Baez
2009-06-05 2:54 John Iskra
2009-06-05 2:42 Hasse Riemann
2009-06-05 1:53 tholen
2009-06-03 20:30 Ronnie Brown
2009-06-03 16:45 Michael Shulman
2009-06-02 16:31 Hasse Riemann
2009-05-23 20:14 Hasse Riemann
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