This is going to be a short but sweet post on topology. Remember way back when I started writing about category theory? I said that the reason for doing that was because it's such a useful tool for talking about other things. Well, today, I'm going to show you a great example of that.
Last friday, I went through a fairly traditional approach to describing the topological product. The traditional approach not *very* difficult, but it's not particularly easy to follow either. The construction isn't really that difficult, but it's not easy to work out just what it all really means.
There is another approach to presenting it using category theory, and to me at least, it makes it a *whole* lot easier to grasp. To make the diagrams easier to draw, I'll adopt one shorthand: instead of writing (T,τ) for topological spaces, I'll use a single symbol, like **X**, with the understanding that **X** represents the *pair* of the set and the topology that form the topological space.
Suppose we have a set topological spaces, **E**1, **E**2, ..., **E**n. The product **P** = Πi=1..n**E**i is the *only* topological space with projection functions pi : **P** → **E**i, such that
for any other topological spaces **S**, if **S** has continuous functions fi : **S** → **E**i to each of the elements of the product, then there is *exactly one* continuous function g : **S** → **P** such that the following diagram commutes:
That's really just a repetition of the definition of categorical product, just made specific to the category **Top**. Everything I said in fridays post about what forms the open sets of the topological product space is directly implied by this categorical definition. The property of the open sets of the product topology being the coarsest structure of sets that maintains the structural properties of the product element topologies - that's implied by the categorical description.
To me, this is the real beauty of category theory, and the whole reason why I spent all that time explaining it. Being able to describe structures in the language of category theory makes things much easier to understand.
I'm a programmer who has built a simple OCR system that might catch your attention. Given that it works well and fast, it puzzles me that I can't find any reference to a similar system. Unlikely as it seems, it might be new.
Some text from http://learning-by-glueing.com/blog
A classifier is a map f from a large topological space X into a small discrete set of symbols denoted Y. We can regard Y as a finite topological space with discrete topology.
In machine learning, sets of examples, e.g. points in X with prescribed images in Y are often called âtraining examplesâ, and many learning algorithms start from them. I strongly prefer the term âreference pointsâ or âreference examplesâ because Iâm convinced that you cannot learn from examples alone. In order to learn anything you have to interact with the world, to move around in X, to explore the spaces of the unknown.
Any example worth mentioning has a small neighborhood in which the class does not change. In other words, it is a germ in the sheaf of locally constant functions f: Xâ¶Y. Any learning algorithm must grow these germs, extend them to large open subsets, large enough to be useful.
My algorithm works by considering several slightly different discretization maps hk from X to small discrete spaces Xk giving a map into Prod(k=1,n,Xk). It's still discrete, but the hamming distance on the product gives a nontrivial notion of neighborhood and covering.
In the OCR example, each map is obtained by cropping a small rectangular patch, scaling down to 6x5 pixels and applying a threshold operation. The small bitmap is used as a memory address. Very slight movements of the patch ( microsakkades? ) will change the address. All these points in Xk are, however, effects of the same cause. They are close together, the classifier should map them all to the same point in Y.
A memory lookup takes a point of Prod(k=1,n,Xk) to Yn. Ideally, it falls into the diagonal Î : Y â¶ Yn. A majority decision provides a map from Ynâ¶Y. Now comes the catch: If majority is high and veto is low, e.g. if image in Yn fits well into the diagonal, we activate a feedback path that writes the majority decision back into the memory cells addressed by the hk(x). The initial germ given by a reference example will grow.
For more information and some graphical examples of hyperacuity, visit http://learning-by-glueing.com/blog .
"The product P = Î i=1..nEi is the only topological space with"
WRONG. There are INFINITELY MANY top. spaces and proj. maps satisfying the definition!! What is true is that the top. prod. is the only topological space UP TO HOMEOMORPHISM satisfying the given property. More precisely, if you define another category whose objects are top. space X with families of morphisms from X to the E_i (the index set need not be finite, btw), and whose morphisms go between top. spaces X and Y respecting the indexed families of morphisms, then the top. prod. is a terminal object in THIS category, and so is defined up to isomorphism.
"That's really just a repetition of the definition of categorical product, just made specific to the category Top. Everything I said in fridays post about what forms the open sets of the topological product space is directly implied by this categorical definition."
Again, WRONG. The fact that products exist in the category of top. spaces is NOT "directly implied by this categorical definition". Otherwise, EVERY category would have products!! You still have to go through what you did on Friday to PROVE that products exist in the category. Definitions don't prove anything.
"The property of the open sets of the product topology being the coarsest structure of sets that maintains the structural properties of the product element topologies - that's implied by the categorical description."
No, it's not "implied by", the categorical description is an abstract embodiment of certain features of the top. product. But the definition "implies" nothing about top. products. (Be careful using the word "implies", it has a precise mathematical meaning.)
"To me, this is the real beauty of category theory, and the whole reason why I spent all that time explaining it. Being able to describe structures in the language of category theory makes things much easier to understand."
Can't argue with you there. I'm still surprised, every time I come here, there's something new to correct.
Yes, I forgot to say "up to homeomorphism"; on the other hand, the rest of your "corrections" are wrong.
For example, the categorical description of the product applied to the category Top *does* imply the necessary properties of topological products. By the strict mathematical definition of "implies".
I'm not going to bother with the rest, because we both know that you aren't *really* interested in whether or not the math here is correct: you're just insulted by the fact that I dared to criticize Duesberg for his incompetent math.
There are actually competent topologists reading my posts, and they *do* frequently correct me, and I do my best to respond to their corrections.