Four problems corrected

["Joyal, André" <joyal.andre@uqam.ca>]

Dear All, 

there was a few typographic errors my last message. 
I am sending you the correction.

Best, 
André


-----------------------------------------------------
 
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 observations 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=SQ:C-->E-->B with Q:C-->E 
a connected Grothendieck fibration and  S:E-->B a discrete fibration 
(the fibers of S 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 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)=colimit D(-,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.
Assuming that the conditions 2 and 3 are 
satisfied, the problem 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.



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