Re: Four problems

[tholen@mathstat.yorku.ca]

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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>




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