Subject: categories: Back to mathematics

From: jean benabou <[email-masked]>
Date: Mon, 9 Jan 2006 10:00:19 +0100

1 - Let P: C ---> B  be a functor. I recall that  V(P) denotes the
subcategory of C having as maps all vertical maps.
I have been working for many years now on the following kind of
questions.
Given an arbitrary subcategory V  of a category C,
(i) - When is it of the form V(P) for some P ?
(ii) - If it is, what relation is there between all the P's such that
V=V(P) ?

2 - Before I go further into these questions, especially (ii) let me
examine a very important special case where complete answers to (i) and
(ii) are well known, namely when C is a group, the answers of course
are:
(i) - When V is a normal subgroup of G
(ii) - If V is normal, let  P: C ---> C/V  be the canonical surjection
on the quotient , and P': C ---> B'  be a functor such  that  V=V(P') ,
(I'm not assuming that B' is a group), then there is a unique functor
Q: C/V ---> B' such that  P'=Q.P , and moreover this Q is faithful .

I apologize for such trivialities but they will permit me to comment on
1-(i)and (ii) and to sharpen them

3 -  The "functor" P of 2 (i) is surjective, but FOR GROUPS this is
equivalent to ;  P is a fibration, which immediately suggests the
following questions for C an arbitrary category and V a subcategory
(i) - If V=V(P)  for SOME P, is there a FIBRATION  P' such that
V=V(P'). If it is not always the case, which V's are of the form V(P)
for a FIBRATION  P ? And of course 1 (ii) can be modified by asking;
what relation is there between all the FIBRATIONS  P which have the
same category V of vertical maps ? This is now purely a question on
fibered categories.
(ii) There are many variations on (i) e.g. replacing  fibration by
prefibration, but even for those who don't like prefibrations, here is
another kind of variation:
A group is a category with pull-backs, and group homomorphisms preserve
pull-backs, so in general one might ask :  If C is a category with
pull-backs, which V's  are of the form V(P) for a functor P WHICH
PRESERVES PULL-BACKS ?

4 - Let us return to groups. If C is a group and V a subgroup, not
necessarily normal, one can denote by  C/V the coset space (say right
cosets) it is of course not a group but inherits a rich structure from
the action of C on it.
Now if C is a category, what does  one need to assume on a subcategory
V of C to be able to construct an analogous C/V and what structure does
it inherit ?

I have studied  many of  he previous questions, and in detail the
question 4 for which I defined prefoliated and foliated categories
which are, roughly speaking, to categorical prefibrations and
fibrations, what topological foliafions are are to fibered spaces. The
quotient  C/V is then a graph, not a category, with extra structure
induced by the action of C, which for obvious reasons I call the
"transverse graph" .

Maybe some persons might consider this as "futile" mathematics. I shall
not try to convince them of the contrary. I'm personally very happy to
do this kind of mathematics, and my motto in this matter, and many
others, is "live and let live"

Greetings to all