Re: localization : more precise question

Re: localization : more precise question

[Philippe Gaucher]

>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 ?

Re: localization : more precise question

[Dan Christensen]

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

> Dan Christensen wrote:
>
> >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. 

...

> 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 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

Re: localization : more precise question

[Philippe Gaucher]

>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.

I would like to add : I meant 'non-negative' locally. Because one needs that the 
morphism from an arrow a-->b to a loop a-->a exists. The exact definition is : 
morphism of local po-spaces (see "Algebraic topology and concurrency", by
Fajstrup, Goubault & Rau{\ss}en ; preprint R-99-2008, Aalborg University).


pg.

Re: localization : more precise question

[Dan Christensen]

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

localization : more precise question

[Philippe Gaucher]

Re-bonjour, 

Thank you for your answers. My question was very general. So here is
the example.

I am going to define the category C and the collection of morphisms S,
with respect to what I would like to localize.

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.


Remark I : in C, an arrow x--> is not isomorphic to a point.

Remark II : an arrow a-->b can be mapped on the loop a-->a with one 
oriented arrow from a to a.


A morphism f of C is in S if and only if f induces an homeomorphism on
the underlying topological spaces. Now here is an example of f\in S
which is not invertible :

a--->b mapped on a-->x-->b 


This morphism has no inverse in C because the image of x must be equal
to a or b by 1) and therefore one of the arrows would be contracted by
2), which contredicts 3).


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.

The Ore condition is not satisfied by S because of this example. The
Ore condition says that for any s:A-->B in S, and any f:X-->B, there
exists t:Y-->X in S and g:Y-->A such that s.g=f.t. Now the
counterexample : A is a--->b, B is a-->x-->b with s as above ; X is
a-->x with the inclusion f from X in B. Then necessarily Y=X and t=Id.
And s.g(x)=b and f.t(x)=x.


Thanks in advance. pg.