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Stream: learning: questions

Topic: continuous functions as functors?


view this post on Zulip David Egolf (Aug 24 2022 at 22:24):

I learned about the concept of "homomorphism" first in the setting of groups. The idea is that they are maps that "preserve the structure". For example, if ϕ\phi is a map between additive groups, then for ϕ\phi to be a homomorphism of additive groups we require ϕ(a+b)=ϕ(a)+ϕ(b)\phi(a+b) = \phi(a) + \phi(b) for all group elements a,ba,b. This condition on preservation of structure is very similar to what we require for functors, to my understanding.

However, I've also learned that continuous maps are the homomorphisms for topological spaces. I was trying to see how we can express that a function is continuous using a similar "structure preserving" requirement as for group homomorphisms.

Let NsR\mathbb{N} \xrightarrow{s} \mathbb{R} be a sequence that converges to SS.
Then, a function ff is continuous at SS if limnf(s(n))=f(S)\lim_{n \to \infty}f(s(n)) = f(S), I think.

Now define a function LL that takes any convergent sequence and returns its limit. Letting CRNC \subseteq \mathbb{R}^\mathbb{N} be the set of real-valued convergent sequences, then L:CRL: C \to \mathbb{R}.
Then L(s)=SL(s) = S. Under this notation, we have that ff is continuous at SS if L(f(s))=f(L(s))L(f(s)) = f(L(s)).
This is similar to the structure-preserving requirement for groups; if we view ++ as a function that takes two inputs, the structure preserving requirement becomes: ϕ(+(a,b))=+(ϕ(a),ϕ(b))\phi(+(a,b)) = +(\phi(a), \phi(b)). It seems like, from this perspective, a continuous functions really is "structure-preserving", where the structure is created by the operation of taking limits. If this is right, it seems cool!

As a sort of follow-up - Is there some way to view continuous functions between topological spaces as functors, from this perspective?

Edit: This was also making me wonder if one could try and define a topological space as a category where you can compute limits of sequences. This led me to discovering this interesting page: https://ncatlab.org/nlab/show/convergence+space . Neat!

view this post on Zulip Zhen Lin Low (Aug 24 2022 at 22:27):

"Homomorphism" and "homeomorphism" are different words...

view this post on Zulip David Egolf (Aug 24 2022 at 22:30):

Zhen Lin Low said:

"Homomorphism" and "homeomorphism" are different words...

Good point, thanks. I will fix it.

view this post on Zulip Max New (Aug 24 2022 at 22:57):

So your definition doesn't work for all topological spaces. The ones where it does work are https://ncatlab.org/nlab/show/subsequential+space

view this post on Zulip David Egolf (Aug 24 2022 at 23:03):

Max New said:

So your definition doesn't work for all topological spaces. The ones where it does work are https://ncatlab.org/nlab/show/subsequential+space

It would be great if you could be a little more specific (which definition? doesn't work how?). I don't really know much topology, and was basically just thinking about real numbers when I wrote this out. But thanks for the interesting link!

view this post on Zulip John Baez (Aug 24 2022 at 23:03):

There are lots of topological spaces where the topology is not determined by the collection of sequences that converge.

view this post on Zulip John Baez (Aug 24 2022 at 23:05):

This is why people invented nets... and I think the link here will help explain the issues.

view this post on Zulip John Baez (Aug 24 2022 at 23:06):

When I teach analysis, I list 3 facts about sequences that are true for metric spaces but not general topological spaces. Later I show how all these facts can be "saved" if we use nets instead of sequences.

The link mentions just one of these facts, under "Motivation". That's the one were just talking about:

1) a function between metric spaces is continuous iff it maps convergent sequences to convergent sequences.

view this post on Zulip John Baez (Aug 24 2022 at 23:08):

Here are the two other facts: for a metric space

2) a subset U is open iff given x \in U and a sequence converging to x, the sequence is eventually in U;

3) a subset K is compact iff every sequence in K has a convergent subsequence.

view this post on Zulip John Baez (Aug 24 2022 at 23:15):

It's good to know how all three of these facts fail for general topological spaces, and how to save them using nets.

view this post on Zulip David Egolf (Aug 24 2022 at 23:17):

Ok, that makes a lot of sense! So in a metric space we have continuous functions interacting nicely with convergent seequences, but that's not the case in every topological space. And apparently this niceness can be recovered by using "nets", which are functions from directed sets (instead of from just the natural numbers).
That explains why my idea above won't work for all topological spaces, at least when working with sequences.
Thanks!

view this post on Zulip John Baez (Aug 24 2022 at 23:18):

Right. Basically the problem is that for really "big" topological spaces, sequences just aren't long enough. They're infinitely long... but that's only the smallest infinity!

view this post on Zulip David Egolf (Aug 24 2022 at 23:21):

Right, our sequences only have countably many elements. But one could consider a map from the ordered real numbers, for example, to get an image with many more points (uncountably many).
Also, I seem to recall that directed sets can have more complicated shapes - some kinds of "branching", I think? I don't know if that is important to the "niceness" of nets.

view this post on Zulip John Baez (Aug 24 2022 at 23:22):

Right. We need to let nets be longer than sequences... but while we're at it, we might as well let them be "wider".

view this post on Zulip John Baez (Aug 24 2022 at 23:23):

That is, we index a net by a directed set, and a directed set doesn't need to be linearly ordered.

view this post on Zulip John Baez (Aug 24 2022 at 23:24):

That lets us do tricks like this. Take as our directed set the collection of all open sets UU containing a point xx, ordered by reverse inclusion so UVU \ge V iff UVU \subseteq V. Then we can take a net indexed by this directed set!

view this post on Zulip John Baez (Aug 24 2022 at 23:26):

This allows really slick proofs.

view this post on Zulip David Egolf (Aug 24 2022 at 23:26):

That's a really cool example! It sounds like that would let you talk about approaching a point from "all sides at once", or without committing to a single path of approach.

view this post on Zulip John Baez (Aug 24 2022 at 23:27):

Exactly! In metric spaces we often say stuff like "choose a sequence xnx_n such that d(xn,x)1/nd(x_n , x) \le 1/n".

But with nets we can say "choose a net xUx_U such that xUx \in U", where UU is any open set containing xx.

view this post on Zulip John Baez (Aug 24 2022 at 23:28):

So once you get used to them they are very easy to work with.

view this post on Zulip John Baez (Aug 24 2022 at 23:29):

The point is that open neighborhoods, rather than distances, are how we measure "closeness" in a topological space.