Famous unsolved problems in ordinary category theory

["Reinhard Boerger" <Reinhard.Boerger@FernUni-Hagen.de>]

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?

Greetings
Reinhard



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