Rafael Borowiecki wrote:
I don't know what to make of the silence to my question. This is the easiest question i have. I can't believe it is so difficult. It is not like i am asking you to solve the problems.
There must be some important open problems in ordinary category theory.
I think the reason for the silence is that category theory is a bit different than other branches of mathematics. Other branches of mathematics get very excited about patterns that may exist, but may not. So when mathematicians hear the phrase "famous unsolved problems", that's the sort of thing that comes to mind: for example, Goldbach's conjecture, the twin prime conjecture, the Riemann hypothesis or the Hodge conjecture. On the other hand, category theorists tend to get excited about taking already partially understood patterns in mathematics and making them very clear. So, the most important open problems often aren't of the form "Is this statement true or false?" Instead, they tend to be a bit more open-ended, like "Develop a workable theory of n-categories." So, they don't have names. I've tried to encourage people to work on n-categories by emphasizing five "hypotheses": the homotopy hypothesis, the stabilization hypothesis, the cobordism hypothesis, the tangle hypothesis, and the generalized tangle hypothesis. I didn't want to call them "conjectures", because they're a bit open-ended. But they're precise enough that someone can claim to have proved one, and people can probably agree on whether this has occurred. For example, Jacob Lurie claims to have proved the cobordism hypothesis: http://arxiv.org/abs/0905.0465 http://lab54.ma.utexas.edu:8080/video/lurie.html and when he provides the full details, people should be able to decide if he has. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]