Four problems

["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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