On Apr 28 2010, Joyal, André wrote:
Jean Benabou has formulated four problems of category theory. They were communicated to a restricted list of peoples, not a private list. I see no serious raisons for not sharing these problems with everyones. Here they are:
Prob1. What conditions must a (small) category C satisfy in order that : there exists a faithful functor F: C --> G where G is a groupoid? (Generalized "Mal'cev" conditions)
This problem is solved in a recent paper of mine "On embedding categories in groupoids" in Math. Proc. Camb. Philos. Soc., vol. 145. Actually, the problem was essentially solved by Mal'cev and (independently) by Jim Lambek, who gave necessary and sufficient conditions for a semigroup to be embeddable in a group; all one has to do is to observe that the same conditions work in the situation when one has several objects. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]