In the article J. Rosicky, W. Tholen, "Factorization, fibration and torsion", J. Homotopy Theory and Related Structures (electronic) 2 (2007) 295-314 we prove a result closely related to Problems 3 and 4 below (variations of which may well have appeared earlier?), as follows: In a finitely complete category C, (E,M) is a simple reflective factorization system of C (in the sense of Cassidy, Hebert, Kelly, J. Austr. Math. Soc 38, (1985)) if, and only if, there exists a prefibration P: C ---> B preserving the terminal object of C with E = P^{-1}(Iso B) and M = {P-cartesian morphisms}. Here "prefibration" means that for all objects c in C, the functors C/c ---> B/Pc induced by P have right adjoints, such that the induced monads are idempotent. (For a fibration one asks the counits to be identity morphisms.) Of course, Jean's question wants P^(-1)(Iso) to be replaced by the non-iso-closed class P^(-1)(Identities), which prevents the class from being part of an ordinary factorization system. But (without having looked into this at all) I would suspect that there is probably a (more cumbersome) reformulation of the theorem above which would address that concern. Regards, Walter. Quoting "Joyal, André" <joyal.andre@uqam.ca>:
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
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