Re: localization : more precise question

[Dan Christensen <jdc@julian.uwo.ca>]

Philippe Gaucher <gaucher@irmasrv1.u-strasbg.fr> writes:

> 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.
> 
> A morphism f of C is in S if and only if f induces an homeomorphism on
> the underlying topological spaces.
> 
> 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.

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