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)
Prob2. A "little" bit harder, in the same vein. Let C be a (small) category, S a set of maps of C and P: C --> C[Inv(S)] be the universal functor which inverts all maps of S. What conditions must the pair (C,S) satisfy so that the functor >P is faithful?
If P: C --> S is a functor, I denote by V(P) the subcategory of C which has the same objects and as maps the vertical maps i.e. the f's such that P(f) is an identity. Let V be a subcategory of a (small) category C. What conditions >must the pair (C,V) satisfy in order that:
Prob3. There exists a functor P with domain C such that V = V(P) Prob4. There exists a fibration P with domain C such that V = V(P)
Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]