Dear Walter, Let me sketch a possible solution to problem 4 (along the lines you have suggested).

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 there exists a fibration P with domain C such that V = V(P)?

I first make a few observation on the properties of a Grothendieck fibration P:C--->B. 1) If V is the subcategory of vertical maps in C, then a morphism in C is cartesian with respect to the functor P iff it is right orthogonal (= it has the unique right lifting property) to every map in V. 2) if R(V) is the class of maps in C which are right orthogonal to every map in V, then every map f in C admits a factorisation f=cv with v in V and c in R(V). 3) the base change of a map in V along a map in R(V) exists and can be taken in V (up to an isomorphism). 4) A Grothendieck fibration P:C--->B is *connected* if its fibers are connected categories. It is easy to see that every Grothendieck fibration P:C--->B admits a factorisation P=DQ:C-->E-->B with Q:C-->E a connected Grothendieck fibration and S:E-->B a discrete fibration (the fibers of D are the connected components of the fibers of P). The vertical maps of P coincide with the vertical maps of Q. This shows that if the problem has a solution, then there is one in which the fibration P:C-->B is connected, in which case there is a natural bijection between the objects of the base category B and the set connected components of the sub-category of vertical maps. We shall suppose the the conditions 1-2-3 above are satisfied but we will we will need an extra condition later. The idea then is to declare that the objects of B *are* the connected components of the subcategory V. If C_0 and C_1 are two connected components of V, consider the distributor D:C_0-->C_1 obtained by putting D(a,b)=C(a,b) for every object a of C_0 and every object b of C_1. The idea is to put B(C_0,C_1)= colimit D and to use the composition of arrows in C for defining the composition of morphisms in B. But we have to make sure that the composition so defined is unambigous. And this is where the extra condition 4 is popping out. First, the distributor D:C_0-->C_1 is locally corepresentable by a family. More precisely, for any object b of C_1 the presheaf D(-,b):C_0-->Set is a coproduct of representable presheaves. This follows directly from condition 2 (the representing objects are morphisms f:a-->b in R(V)). Let us put T(b)=\pi_0D(-,b) This defines a functor T:C_1--->Set (which depends on C_0). It follows from condition 3 that the functor T inverts every morphism of C_1. This shows that the distribuor D:C_0-->C_1 is of a very special type. The category of elements of the functor T is a *covering* el(T)-->C_1 (a covering is a discrete fibration which is also an opfibration). It follows that the distributor D:C_0-->C_1 can be represented as a span C_0<---el(T)--->C_1 in which the second leg is a covering. The problem 4 of Benabou will have a solution iff this covering is trivial (ie it is a product) for any pair of connected components C_0 and C_1 of V. This is true for example when the connected components of V are simply connected. Best, André -------- Message d'origine-------- De: tholen@mathstat.yorku.ca [mailto:tholen@mathstat.yorku.ca] Date: ven. 30/04/2010 21:13 À: Joyal, André Cc: categories@mta.ca; tholen@mathstat.yorku.ca Objet : Re: categories: Four problems 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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]