Duality between metric and logical viewpoints

[anhinga_anhinga]

The metric viewpoint: how far two objects are from each other. The logical viewpoint: to what degree two objects overlap.

Fuzzy mathematics is traditionally done from the logical viewpoint, so the first step in introducing fuzzy metrics is often the transformation f(x,y) = exp(-d(x,y)).

Then we have the following correspondences:

d(x,y) = 0 if and only if f(x,y) = 1.

d(x,y) is plus infinity if and only if f(x,y) = 0.

d(x1, y1) < d(x2, y2) if and only if f(x1, y1) > f(x2, y2).

The axiom d(x,x) = 0 becomes f(x,x) = 1.

The axiom d(x,z) < d(x,y) + d(y,z) becomes f(x,y) * f(y,z) < f(x,z).

Non-expansive maps become maps which respect overlap by not letting it decrease.

Etc..


However, this seems to be a rather superficial duality: basically two equivalent ways to write the same things using different notation.

The question is whether there is also a natural deeper duality here (of a contravariant nature, where function arrows would reverse direction when one switches between these two viewpoints).

Comments

[algebraic-brain.livejournal.com]

Very good question I think. Thank you.

[dklein.livejournal.com]

I don't think there is :(

[anhinga-anhinga.livejournal.com]

I also cannot find one so far, despite various hints I am seeing that there might be one :(

Can you elaborate why your intuition tells you there is no contravariant duality here?

******************

Among the hints I am seeing is that if we consider numbers in the metric approach (aka truth values in the logical approach) as a category (by their linear or, in general, partial order), then 0 is the initial object, but its logical equivalent, 1 (the "truth"), is the terminal object.

Also this exp going from metric to logical (and log going back), and the associated switches between additive and multiplicative notation (the use of + is typical for the metric viewpoint in general, and the use of * is typical for the logical viewpoint in general) might be less than completely superficial. In the simplest case, where + is just max (sup), and * is just the conjunction (inf, min), a+b is really the coproduct, and a*b is the product in the category of numbers (truth values).

But, of course, all these hints are about numbers/truth values themselves, not about the associates metrics/"fuzzy metrics" ("fuzzy equalities" might be a better term here, which f(x,y) * f(y,z) < f(x,z) being essentially a transitivity axiom). And they come from simply flipping the order. So none of these hints is sufficiently conclusive to suggest there ought to be a good contravariant duality between metrics and fuzzy equalities.

[anhinga-anhinga.livejournal.com]

Please let me know, if you can think of something, or of some analogies in the literature.

(See also the next comment thread.)

[cema.livejournal.com]

I still do not see much I could add to what you said. One thing perhaps: maybe you could list typical applications (for each viewpoint) and try to express it in terms of the other viewpoint. Maybe it will give you some insight, if not in terms of the relation between the view points, then in terms of the application area.

[dklein.livejournal.com]

Because the functor you are considering, from (X,d) to (X,f),
is just the identity map on X, why would it reverse the arrows?
or am I missing something?

>if we consider numbers in the metric approach (aka truth values in the logical approach) as a category
This I don't understand, which numbers? reals? what are morphisms in this category?

[anhinga-anhinga.livejournal.com]

> the identity map on X

This functor would not reverse the arrows. This particular functor provides "superficial duality".

But there are various interesting functors within logical viewpoint (and within metric viewpoint). Perhaps, one of them should really be understood as a "functor between these viewpoints".

For example, there is a cute construction of "singleton completion", which is one of the ways to do "sheafification". Here one fixes a point x, and considers the function mapping y to f(x,y). This function has certain properties, when one axiomatizes them one gets so-called "singletons", functions which could potentially have been mapping y to f(x,y) for some x. The space of such singletons is the "singleton completion", and that's how one goes from a set equipped with a "fuzzy equality" valued in a complete Heyting algebra to a sheaf over that complete Heyting algebra (a segment of real numbers is a slightly degenerate example of complete Heyting algebra). Now, this sheafification is a covariant functor, although it is "morally" an embedding from a set to its dual (given that singletons are functions from this set to the truth values in question). But this situation is one of the possible contexts in which we can try to think about nontrivial duality.

> reals? what are morphisms in this category?

For example, reals. Just like with any partial order, we construct a category by saying that two points are connected with a single arrow from x to y if and only if x is less or equal to y.

Some of these number systems (or systems of truth values) are linearly ordered, some are partially ordered (like Boolean or Heyting algebras), but all can be made into categories by this construction.

[anhinga-anhinga.livejournal.com]

That's a possibility.

:-) Of course, what I was mostly doing with this was taking the (superficial) duality I understood and using it to get sometimes non-trivial/unexpected links between various applications :-)

But if I were able to come up with the links from some other considerations, this could have potentially provided some insight, yes..

[dklein.livejournal.com]

>This functor would not reverse the arrows
that's what I meant :)

>construct a category by saying that two points are connected with
>a single arrow from x to y if and only if x is less or equal to y.
OK, then any decreasing map from R to itself will reverse arrows :)

[anhinga-anhinga.livejournal.com]

> OK, then any decreasing map from R to itself will reverse arrows :)

In a category of numbers, yes. But the question was about various categories of metric and metric-like spaces with values in those numbers.

One remark is that a formal reversal of arrows is easy to achieve: one just starts with any covariant functor and replaces one of the categories with its dual. The question is not about whether something like this can be done, but whether there are naturally occurring dualities here.

[dklein.livejournal.com]

For metric spaces, I think I explained why I don't see any non-superficial duality :)

[anhinga-anhinga.livejournal.com]

I don't think you did :-)

As I explained above, one of the many places to look for such duality is to consider a certain class of functions from a metric space to numbers in question. These functions are usually studied from the logical viewpoint and are called singletons there :-)

You certainly explained why you did not see any non-superficial duality looking at the functor which is the identity on the class of objects, but I don't think you explained why there should not be a non-superficial duality in the less trivial setups, for example, between a space and the associated space of singletons.

[dklein.livejournal.com]

OK, let me try again. Essentially the above discussion showed that
your superficial-duality functor establishes an equivalence between
the category with objects (X,d) and the one with objects (X,f).
Certainly there exist covariant functors, like the one you described,
but the aforementioned equivalence shows that they are in 1-1 correspondence
with functors from the category of usual metric spaces to itself :)

[anhinga-anhinga.livejournal.com]

Sure, but there must be plenty of dualities within that category (just like we have dualities within the category of vector spaces), including the ones which are unknown, or considered unremarkable.

(One should note the number of various interesting categories of metric and metric-like spaces is quite high, because there are many attractive variants for the notion of morphism. So we really are talking about a family of equivalences, and in some cases the resulting notions of morphism are new for the metric viewpoint, or for the logical viewpoint.)

*************

You are certainly correct that any duality which might be found here would allow reformulation as a duality within metric viewpoint (although it might become less cute in the process).

I think the question whether interesting dualities can be obtained from considering the interplay between metric and logical viewpoints remains quite valid.

[anhinga-anhinga.livejournal.com]

> I think the question whether interesting dualities can be obtained from considering the interplay between metric and logical viewpoints remains quite valid.

But it is correct that this anything obtained along these lines, in principle can be obtained while staying entirely within one of these viewpoints, with enough brainpower..

[shamebear.livejournal.com]

Mathoverflow.net seems to be the new in-place for math. I suspect a Prof. Tao's advertising of it may have played a part in that. :-) Anyway, when I search for "fuzzy categorification" there are several threads on categorifying fuzzy math. I think that taking the duality question into the grand categorification project might be fruitful. "Deeper connections" seems to be that project's slogan :-)

There's also a thread on linking euclidean and combinatorial distance

[anhinga-anhinga.livejournal.com]

Thanks!