Re: localization : more precise question

[Philippe Gaucher <gaucher@irmasrv1.u-strasbg.fr>]

>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 ?