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.
I believe that you are wrong somewhere. The explanation is in post-scriptum (borrowed from a question in sci.math.research which is not yet posted by now). Or maybe I am wrong in the reasonning ?
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?
I meant 'non-negative'. Maybe the definition of the category still needs to be debugged. I don't know. (The motivation of this question was to encode the notion of 1-dimensional HDA up to dihomotopy for those who know the subject in a "true" category such that isomorphism classes represent 1-dimensional HDA up to dihomotopy). "having a 'positive' derivative at all times" would be also sufficient I think. Cheers. pg. PS : The natural conjecture is that C[S^{-1}] is equivalent to the category D 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. If U is a universe containing all sets, let V be a universe with U\in V. The categorical construction of C[S^{-1}] (let us call it "C[S^{-1}]") gives a V-small category. "C[S^{-1}]"(A,B) is generally not a set. To see that, take an object like g_1.f_1^{-1}.g_2 with g_1 and g_2 not invertible in C (this is a reduced form which cannot be simplified in "C[S^{-1}]"). Then replace f_1^{-1} by f_1^{-1} \sqcup Id_{codom(f_1^{-1})} \sqcup Id_{codom(f_1^{-1})} \sqcup... and g_1 by g_1 \sqcup g_1 \sqcup g_1 ... Then "C[S^{-1}]"(dom(g_2),codom(g_1)) has as many elements as the sets of U-small cardinals. Therefore "C[S^{-1}]"(dom(g_1),codom(g_2)) is not a set. The relation between "C[S^{-1}]" and D is as follows. There is a canonical V-small map g : "C[S^{-1}]"(A,B) --> Sets(A,B) and D(A,B) is the quotient of the V-small set "C[S^{-1}]"(A,B) by the V-small equivalence relation "x equivalent to y iff g(x)=g(y)". The above element of "C[S^{-1}]"(dom(g_2),codom(g_1)) are all of them identified by this equivalence relation : it is the reason why the homset from dom(g_2) to codom(g_1) becomes a set. The obvious functor from C-->D does invert the morphisms of S. But one has to prove that for any functor C-->E inverting the morphisms of S, C-->E factorizes through C-->D by a unique functor from D-->E. Such functor C-->E factorizes through "C[S^{-1}]" but for proving the factorization through D, one has to prove that E is a sort of concrete category (a category with a faithful functor to Sets). Of course there is no reason for E to be concrete but because of the functor F:C-->E, Im(F) is not too far from a concrete category. C is a concrete category, constructed with oriented graphs. I never heard about a general way of constructing localizations of concrete categories. Does it exist ?