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Stream: learning: questions

Topic: Contextualizing how continuity is taught with categories

Alex Kreitzberg (Aug 30 2025 at 19:55):

When learning calculus rigorously, in my head I binned all the functions into a "platonic universe" organized by properties, like in the following diagram:

cOdFy.png

Learning categories proper has kind of broken this image in my mind though. Because, for example, the limits of a category aren't preserved under subset restriction.

So when a function is defined as continuous by using some metrical structure, I want to say there is a functor in the background relating this to the implicit topology induced by that metric.

It looks like I'm adding a property to restrict the subset the function belongs to, but really I think there's a forgetful functor from a category whose morphisms are continuous metrical functions into the category of topological spaces. I feel clumsy doing this at the moment.

The "set theory with subsets" viewpoint makes it seem easy to move an imagined "platonic" function between the various subsets. Maybe for a given function $f$ I'll prove a fixed point exists by proving it's continuous, and then find the solution by proving the function is differentiability. But, presumably moving between levels like this can get one into trouble, because we're carelessly switching categories.

When does the subset picture indicated by the above diagram get me into trouble? And how do I fix it using a categorical perspective? How do I navigate the various categories conveniently in practice?

A bit ago on this zulip, I was encouraged to imagine a sphere as existing in multiple distinct categories, each with their own theorems. In particular I'm trying to not interpret the sphere as a "platonic" set with distinct properties including it in various sets of geometric subsets. The above questions could be interpreted as me trying to relate my old, but I think classical, understanding of calculus and euclidean geometry with this new point of view made possible by category theory.

I guess to expand briefly on the "euclidean geometry" point of that last sentence. My classical understanding was, when Euclid says "draw a line between two points" I should think of the line as existing in the real world, that is geometry was a science. If I wanted it to be ideal when the approximation failed, I could instead talk about a "platonic" ideal line in the euclidean plane, imagined as a set of points with a metrical structure. My understanding of the modern point of view, is it's very easy to approximate a given real world thing with many different possible mathematical structures (a set, a linear relation, a parametric function, etc.), or that a given mathematical description of something is necessarily abstract. And again, I'd appreciate advice navigating the "classical" and "modern" viewpoints.

Alex Kreitzberg (Aug 30 2025 at 20:21):

Maybe an aside that's close to this in my brain:

Awhile ago I tried to make a "category for drawing", but I decided with how I was understanding drawing, in a certain sense I was just reconstructing all the various "space like"/"geometric" categories. And in particular a given line I might draw, could be easily imagined as approximable by lots of different categories. This is still a bit disorienting to me, I'm trying to get more of a handle on this with the above questions.

John Baez (Aug 31 2025 at 09:20):

When does the subset picture indicated by the above diagram get me into trouble?

That diagram doesn't mean much to me: it looks like a bunch of boxes with numbers and calligraphic letters in them, but no accompanying explanation except "The set of all functions". Are we supposed to understand it in some detail, or it just an example of how the set of all functions has subsets?

Alex Kreitzberg (Aug 31 2025 at 13:52):

Sorry, its key got dropped:

1000020515

Alex Kreitzberg (Aug 31 2025 at 14:10):

It's from this book:

https://assets.cambridge.org/97805212/83618/excerpt/9780521283618_excerpt.pdf

John Baez (Aug 31 2025 at 14:52):

Okay! They much be talking about functions from $\mathbb{R}$ to $\mathbb{R}$. The real line is a somewhat unusual entity in that it's typically defined as a [[complete ordered field]], and when we give it all that structure, it's not only uniquely characterized up to isomorphisms - it has no endomorphisms except the identity. In hand-wavy terms, it's completely rigid.

So we have to remove structure from that complete ordered field to make it more floppy - to give it more endomorphisms.

This is different from complex numbers, where you've always got complex conjugation until you say which number is $i$ and which is $-i$. It's dramatically different from the field of algebraic numbers, which is cursed with a plague of automorphisms.

Alex Kreitzberg (Aug 31 2025 at 15:29):

I'm getting the sense from your answer, that my understanding for how functions on the real line are related won't generalize to how functions defined on other spaces work.

So if I look at the homset $\text{Set}(\mathcal{R}, \mathcal{R})$ of functions between real numbers, there are no nontrivial isomorphisms relating this to some other homset $\text{Set}(X, Y)$. And therefore when understanding real functions with special properties, it amounts to choosing subsets of $\text{Set}(\mathcal{R}, \mathcal{R})$.

When nontrivial automorphisms enter the picture this is no longer true, now we start having categorical flavored relationships operating at a higher level than just the subsets.

Is this a good way to understand the significance of your examples?

John Baez (Aug 31 2025 at 17:34):

So if I look at the homset $\text{Set}(\mathbb{R}, \mathbb{R})$ of functions between real numbers, there are no nontrivial isomorphisms relating this to some other homset $\text{Set}(X, Y)$.

Huh, what? That barely makes sense. But more importantly, it sounds false.

In the category of sets and functions there are tons of automorphisms of $\mathbb{R}$: any bijective map sending real numbers to real numbers will do. There are even more isomorphisms between the set $\text{Set}(\mathbb{R}, \mathbb{R})$ and certain other homsets $\text{Set}(X, Y)$. If I count correctly, there can be exactly

$2^{2^{\aleph_0}}$

of them.

What I said is that usually $\mathbb{R}$ is given as a complete ordered field. Then people prove that given any two complete ordered fields, there's exactly one morphism between them.

So at this point, at its birth, $\mathbb{R}$ is not a mere set. It's a complete ordered field. It's been given so much structure that it has no nontrivial automorphisms - or even endomorphisms. We then spend a lot of time 'stripping off' or 'forgetting' structure: hitting it with functors that turn it into a topological space, or a manifold, or an ordered set, or whatever. This makes it more flexible.

Alex Kreitzberg (Aug 31 2025 at 18:43):

Okay, you're touching on something I'm deeply confused about, which is good. I'll do my best to try to ask questions that give some sense of where my head is.

I wanted to define operations on the complete ordered field, but I can't because it has so much structure that in order to preserve it I'm stuck with the identity function. So like you said we forget structure to give us room to do more things.

But often it seems like when I'm working with the complete ordered field $\mathcal{R}$, I forget everything, but keep the forgotten structure in the back of my mind, and add it back when I can.

So given functions $f, g : \mathcal{R} \rightarrow \mathcal{R}$, with $f(x) = x^2$ I know it's differentiable, but with $g(x) = x^3$ It's even monotonic! If I have an absolute value function it's merely continuous. I'm choosing functions, and noting later the structure that hasn't been forgotten. But they are all written as if they're functions from $\mathcal{R} \rightarrow \mathcal{R}$, arrows in $Set$. They all are living together in an analysis textbook, but "actually" live in different categories.

I have a feeling or intuition that I'm confused, because if I'm working in $\text{Set}$, like you said, at any time I could compose whatever composite function I'm building with an arbitrary function thereby completely scrambling up what I was doing.

But, thinking I'm working in $\text{Set}$, is how I've been pretending to coordinate functions from different categories. I imagine what their graphs look like, decide what properties I think they have, and move forward from there.

How is this coordination actually supposed work?

Alex Kreitzberg (Aug 31 2025 at 18:48):

Some other aspect that's confused me, that I see connected with this somehow. I can visualize any function $f : \mathcal{R} \rightarrow \mathcal{R}$ by using the metric on $\mathcal{R}$, but most of the functions don't preserve distance, so in what sense is it meaningful to graph these functions?

John Baez (Aug 31 2025 at 18:56):

Math rarely uses just one category at a time. Note that the very definition of a field starts by saying "it's a set with..." a bunch of operations that are functions between sets. So the category of fields, like many categories, is defined as a category with af forgetful functor to Set, and you can't do anything (or least not much) without using that.

Most mathematicians work promiscuously among many categories and are wise enough to not pay too attention to this. If you do, you could easily become like millipede who tries to keep track of its legs and gets paralyzed! But you can figure it out if necessary.

Alex Kreitzberg (Aug 31 2025 at 20:02):

Maybe we could focus on a characteristic or simple example so I can get a feeling for how this works?

Like what were you trying to illustrate by pointing out the category of complete ordered fields had only one object up to unique isomorphism, whereas other algebraic categories have several? It seems they all use forgetful functors into $\text{Set}$ in an analogous way. In what way are the automorphisms of the field of algebraic numbers a plague?

I'd think more automorphisms would be useful exactly because it makes the object less rigid, but instead it seems that's what the forgetful functors are for.

Alex Kreitzberg (Aug 31 2025 at 21:28):

(I'm trying to fix my confusion that caused the $\text{Set}(\mathcal{R}, \mathcal{R})$ mistake, I'm failing to juggle the categories now that I know they're there :joy:)

John Baez (Aug 31 2025 at 22:01):

Alex Kreitzberg said:

Like what were you trying to illustrate by pointing out the category of complete ordered fields had only one object up to unique isomorphism, whereas other algebraic categories have several?

I was mainly saying that since all the morphisms in the category of complete ordered field are isomorphisms, and there's a unique isomorphism between any two complete ordered fieldd, you're doomed if you're hoping for any interesting functions from the real numbers to themselves to be morphisms in the category of complete ordered fields. So despite how we say "the real numbers are defined to be a complete ordered field", we need to go down to some other category (like Set, or Top, etc.) to find interesting functions from the real numbers to themselves.

On the other hand, this rigidity means that we can call any complete algebraic field "the" real numbers, and never get in trouble: any one of your real numbers corresponds to a unique one of mine.

In what way are the automorphisms of the field of algebraic numbers a plague?

I was just joking, but it's well-known that the group of automorphisms of the field of algebraic numbers is huge and unknowable, and there's an apocryphal story that thinking about it has driven mathematicians mad. I've never heard of any evidence for that story. Grothendieck did work on that group (cf. Grothendieck-Teichmuller theory) and he did sort of go mad, but probably not because of trying to understand that group.

Alex Kreitzberg (Aug 31 2025 at 22:09):

So, if I imagine the category of complete ordered fields, it's just a point right? And then we have forgetful functors from that point to other categories, that all also have faithful forgetful functors into $\text{Set}$?

So do we have a sort of lattice of forgetful functors? Is this related to the subset picture I shared at the beginning?

Alex Kreitzberg (Aug 31 2025 at 22:21):

The point would pick out an object in each of these categories, and if all the functors were faithful, you could float up your maps in $\text{Set}$ provided the functions you're lifting can be equipped with the forgotten structure from $\mathcal{R}$ safely, which is what it means to be a faithful forgetful functor because of [[stuff, structure, property]]

I think that makes sense

Alex Kreitzberg (Aug 31 2025 at 23:04):

Can I even save my earlier confused $\text{Set}$ statement?

The category of complete ordered fields has one object up to unique isomorphism, which I'll label $\mathcal{R}$, and therefore one homset $\text{COF}(\mathcal{R}, \mathcal{R})$ with just the identity.

A faithful functor from this into $\text{Top}$ is given by an injective function from $\text{COF}(\mathcal{R}, \mathcal{R})$ to $\text{Top}(\mathcal{R}, \mathcal{R})$.

Then the usual forgetful functor from topological spaces into $\text{Set}$ has an injection from $\text{Top}(\mathcal{R}, \mathcal{R})$ into $\text{Set}(\mathcal{R}, \mathcal{R})$ where $\mathcal{R}$ in each of these cases are what the "point" of the category of complete ordered fields maps to under the forgetful functors into $\text{Top}$ and $\text{Set}$.

So in this way, we get nested injections ending in $\text{Set}(\mathcal{R}, \mathcal{R})$ which is the same as a diagram of subsets.

So I think we do have a meaningful relationship between these forgetful functors and the illustration of subsets I shared at the beginning of this thread.

John Baez (Sep 01 2025 at 08:54):

I think what you're saying is perfectly correct now! :thumbs_up: