Subject: categories: localization : more precise question

From: Philippe Gaucher <[email-masked]>
Date: Fri, 1 Dec 2000 22:13:06 +0100 (MET)

Re-bonjour, 

Thank you for your answers. My question was very general. So here is
the example.

I am going to define the category C and the collection of morphisms S,
with respect to what I would like to localize.

The object of C are the oriented graph. Such an object X is a
topological space obtained by choosing a discrete set X^0 and by
attaching 1-dimensional cells *with orientations*. It is a
1-dimensional CW-complex with oriented arrows.

The morphisms of C are the continuous maps f from X to Y satisfying
this conditions :

1) f(X^0)\subset Y^0

2) f is orientation-preserving

3) f is non-contracting in the sense that a 1-cell is never contracted
to one point.

Remark I : in C, an arrow x--> is not isomorphic to a point.

Remark II : an arrow a-->b can be mapped on the loop a-->a with one 
oriented arrow from a to a.

A morphism f of C is in S if and only if f induces an homeomorphism on
the underlying topological spaces. Now here is an example of f\in S
which is not invertible :

a--->b mapped on a-->x-->b 

This morphism has no inverse in C because the image of x must be equal
to a or b by 1) and therefore one of the arrows would be contracted by
2), which contredicts 3).

I would like to know if C[S^{-1}] exists or no (in the same universe).

The irresistible conjecture is of course that C[S^{-1}] is equivalent
to the category whose objects are that of C and whose morphisms from A
to B are the subset of C^0(A,B) (the set of continuous maps from A to
B) containing all composites of the form
g_1.f_1^{-1}.g_2...g_n.f_n^{-1}.g_{n+1} where g_1,...,g_{n+1}, are
morphisms of C and f_1,...,f_n morphisms in S.

The Ore condition is not satisfied by S because of this example. The
Ore condition says that for any s:A-->B in S, and any f:X-->B, there
exists t:Y-->X in S and g:Y-->A such that s.g=f.t. Now the
counterexample : A is a--->b, B is a-->x-->b with s as above ; X is
a-->x with the inclusion f from X in B. Then necessarily Y=X and t=Id.
And s.g(x)=b and f.t(x)=x.

Thanks in advance. pg.

---

From: Dan Christensen <[email-masked]>
Date: 02 Dec 2000 13:05:01 -0500

Philippe Gaucher <[email-masked]> writes:

> the underlying topological spaces.
> 

Call the category you describe D.  There is an obvious functor C --> D
which inverts the morphisms of S.  So there is an induced functor
C[S^{-1}] --> D, which is the identity on objects and is clearly full,
since the morphisms from A to B in C[S^{-1}] can be described as the
*formal* composites g_1.f_1^{-1}.....f_n^{-1}.g_{n+1} (modulo certain
relations), where g_1,...,g_{n+1}, are morphisms of C and f_1,...,f_n
morphisms in S.

The question is whether the functor C[S^{-1}] --> D is faithful.

I suspect that this is true in general, but can only prove it if
you restrict yourself to CW-complexes with a finite number of cells.
If you do this, then I believe that the reverse Ore condition holds:
given
        s
     A ---> B
     |
     v
     C

with s in S, there exists

        s
     A ---> B
     |      |
     v      v
     C ---> D
        t

with t in S.  (B is just A with a finite number of vertices added;
just add the images of those points in C as new vertices to get D.)

With this, it isn't hard to see that the functor is faithful.

For infinite CW-complexes, this Ore condition doesn't hold, but I
still suspect that the functor is faithful.  In part it depends upon
what you mean by "orientation preserving".  Does this mean "having a
'positive' derivative at all times"?  Or 'non-negative'?  Or can the
map go forwards and backwards as long as overall it has degree one?

Dan