localization : more precise question
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.
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
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 ?
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.
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
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