Subject: categories: Re: localization : more precise question

From: Philippe Gaucher <[email-masked]>
Date: Sun, 3 Dec 2000 00:33:28 +0100 (MET)

>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 ?

---

From: Dan Christensen <[email-masked]>
Date: 03 Dec 2000 17:39:20 -0500

Philippe Gaucher <[email-masked]> writes:

> Dan Christensen wrote:
> I believe that you are wrong somewhere. 

...

> of U-small cardinals. Therefore "C[S^{-1}]"(dom(g_1),codom(g_2)) is

The maps you've described all represent the same map in C[S^{-1}].
For example, the diagram

    g_1        f_1           g_2
  A ---> C <---------- D ------------> B
  |      |             |               |
  |1     |             |               |1
  v      v             v               v
    g_1        f_1 v 1       g_2 v g_2
  A ---> C v D <------ D v D --------> B

shows that the top map is equal to the bottom map in C[S^{-1}].

So this doesn't seems to be a counterexample to my guess that 
C[S^{-1}] exists in general.

(By the way, my argument for the case when the CW complexes are
finite, while possibly useful as an idea for how to approach the
general case, is more complicated than is necessary, since with
this assumption C is equivalent to a small category, and so any
localization of C exists.)

Dan