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