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
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