Topological Products Redux: Categories to the rescue!

Sep 25 2006 Published by under Category Theory, topology

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:

product.jpg

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.

No responses yet

  • Darin Brown says:

    "The product P = Πi=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.

  • Mark C. Chu-Carroll says:

    Darin:
    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.