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Stream: learning: questions

Topic: [0, 1] → X vs X → [0, 1]

Alex Kreitzberg (Apr 19 2025 at 05:43):

Technical art uses parameteric equations and scalar fields. Depending on which is easier for the problem to be solved.

I like to imagine parametric equations as roughly modeling drawing, and scalar fields as roughly modeling painting.

When you give explicit function signatures, it's clear these concepts are dual.

There's lots of flexibility in our choices, but a representative example of this duality is

$[0, 1] \rightarrow X \leftrightsquigarrow X \rightarrow [0, 1]$

Where $X$ is some topological space.

In my mind an example of a function on the left can be visualized as a "drawing":

c218a217-cb9f-418c-bfc2-e5a94b97e7f8.png

And a function on the right can be visualized as a "painting":

5224d3a7-01c5-4c62-b67e-ee639be2f6d4.png

If $X = [0, 1]^2$ then you'd be drawing or painting on a square paper.

My intuition, is "drawings" and "paintings" are related. You can make a map of countries by drawing their borders, or if you have at least four colors you can paint a map of countries.

Irrespective of my intuition. This duality seems to be really important in category theory. The "drawings" correspond to Lawvere's notion of "figures", the "paintings" correspond to his notion of "cofigures".

They're both examples of representable Functors.

In [[space and quantity]], if I'm understanding it right, what I call a set of "paintings" is a presheaf on topological spaces, or a "generalized space".

And what I call a set of "drawings" is a copresheaf, or a generalized quantity.

Whether you think of these as scalar fields vs parametric functions or "paintings" vs "drawings", these dual perspectives are clearly closely related, even mutually convertible.

Where should I start, or what should I think about, to get a better understanding of these categorical definitions and their duality? Are presheafs and copresheafs a good place to start for example?

John Baez (Apr 19 2025 at 06:16):

In physics I like to think of a function $\mathbb{R} \rightarrow X$ as a 'particle' and a function $X \rightarrow \mathbb{R}$ as a 'field', where $X$ is 'spacetime'. Particles and fields give dual ways of thinking about matter. I'd never noticed that particles are like drawings and fields are like paintings. Nice!

A space $X$ with a function $[0,1] \to X$ is called a space 'under' $[0,1]$, and the category of all spaces under $[0,1]$ is an example of an 'under category'.

A space $X$ with a function $X \rightarrow [0,1]$ is called a space 'over' $[0,1]$, and the category of all spaces over $[0,1]$ is an example of an 'over category'.

So one way to learn more is to learn about properties of [[over categories]] (also called 'slice categories') and [[under categories]] (also called 'coslice categories').

Lawvere wanted 'spaces', of a very general sort, to be objects of a topos. Then there's a nice theorem that the over category of any object in a topos is again a topos. This is often called the fundamental theorem of topos theory.

It doesn't work like this for under categories! The under category of an object in a topos is not usually a topos. So there's a built in asymmetry in the concept of topos (and other aspects of our concepts of space). If your category of spaces $X$ is a topos, then the category of spaces over $[0,1]$ (spaces $X$ equipped with a map $X \to [0,1]$) is a topos, but the category of spaces under $[0,1]$ (spaces $X$ with a map $[0,1] \to X$) is usually not.

So, if you set things up nicely, you can get a topos of paintings, but maybe not a topos of drawings.

Mike Shulman (Apr 19 2025 at 06:39):

However, if you are willing to discard the requirement that a drawing sits on a pre-existing "canvas", then you can get a topos of "abstract drawings", namely the topos of (pre)sheaves on the category of "drawable shapes".

Alex Kreitzberg (Apr 19 2025 at 07:04):

I don't normally think of a mathematician's chalk line drawings as being more limited than shaded chalk "paintings", but ya'll have got me thinking they may be in some subtle way.

Shulman I'm extremely curious about your comment, but I don't quite understand it. What's the category of "drawable shapes"? Did I implicitly define that?

Alex Kreitzberg (Apr 19 2025 at 07:06):

Is it the under category $[0, 1] \rightarrow X$?

Alex Kreitzberg (Apr 19 2025 at 07:08):

(And I'm resisting the urge to ask Baez about particle wave duality here, these answers are pulling my brain in a million directions, I'm working hard to restrain myself :joy:)

John Baez (Apr 19 2025 at 20:46):

Alex Kreitzberg said:

Is it the under category $[0, 1] \rightarrow X$?

Note that $[0,1] \to X$ is not an under category! The under category of $[0,1]$ is a category with all maps $f: [0,1] \to X$ for all spaces $X$ as objects.

Puzzle. The morphisms in the under category $X$ are the 'obvious things'. What are they?

Alex Kreitzberg (Apr 19 2025 at 20:51):

A morphism between $g: [0, 1] \rightarrow X$ and $h: [0, 1] \rightarrow Y$ is a function $f : X \rightarrow Y$ such that

$f \circ g = h$

Alex Kreitzberg (Apr 19 2025 at 20:54):

So a (maybe silly) example of an $f$ would be one that bends flat paper into a cylinder. Then $h$ is the drawing you get on the cylinder after bending the drawing you made on the flat paper with $g$.

John Baez (Apr 19 2025 at 21:07):

Exactly! Or you could have a 3d "drawing" and then project it down to two dimensions.

John Baez (Apr 19 2025 at 21:09):

I don't normally think of a mathematician's chalk line drawings as being more limited than shaded chalk "paintings", but ya'll have got me thinking they may be in some subtle way.

It would be wonderful if this mathematical fact - that the over category of an object in a topos is always a topos, but not the under category - was the reason why art museums tend to have a lot more paintings than drawings. But I kinda doubt it.

Alex Kreitzberg (Apr 19 2025 at 21:20):

I doubt it too!

I'm more confused what verbs scaler fields could have that parametric functions apparently lack.

Alex Kreitzberg (Apr 19 2025 at 21:51):

Assuming I understand what it means to be a topos, there's a mystery (to me) here.

Alex Kreitzberg (Apr 19 2025 at 22:11):

And then Shulman's observation reads to me as suggesting if we replace what I called drawings by something slightly more abstract, we get the missing verbs back.

It's all very dramatic and exciting, I wish I understood it better :joy:, perhaps that would diffuse the excitement.

John Baez (Apr 19 2025 at 23:01):

I recommend reading those nLab articles on the properties of over and under categories, for starters...

Mike Shulman (Apr 19 2025 at 23:06):

If the only things you're drawing are intervals [0,1], then the category of "drawable shapes" will have just one object, [0,1]. Its morphisms will be whatever "sub-intervals" and "reparametrizations" you want to allow on your drawn intervals.

In the simplest case there are none (except the identity) and the category of drawable shapes is the [[point]]. In this case, an "abstract drawing" consists of nothing but a set of "drawn intervals". There's no "canvas" on which they're drawn, they're just "abstractly drawn" in space and have no relation to each other. So in this case the category of "abstract drawings" is equivalent to Set.

Of course you may want to be able to do things like "draw a circle". How can we draw a circle with intervals? We can draw two intervals as long as they match up at the endpoints. If our category of drawable shapes includes the two functions $[0,1] \to [0,1]$ that are constant at the endpoints, then we can "draw a circle" abstractly by saying that there are two "drawn intervals", say $\alpha$ and $\beta$, such that if you restrict $\alpha$ along the constant-at-0 function to get a "constant drawn interval", you get the same thing as if you did that to $\beta$, and likewise for the constant-at-1 functions. So we can "draw a circle" abstractly without having any canvas on which to have drawn it.

If you take this to an extreme, you can let the morphisms in your category of drawable shapes to be all continuous functions $[0,1]\to [0,1]$, or all smooth functions. Eventually this direction leads to fancy things like [[smooth sets]].

Alex Kreitzberg (Apr 20 2025 at 04:15):

A glance at the nLab articles indicates that apparently you can do a ton more with over categories than with under categories. I'll try to give some concrete examples of what that means for "paintings"

But for now, let's see if I can follow Shulman's outline.

$PSh(C) := [C^{op}, \text{Set}]$ is the category of Functors from the opposite category of $C$ into $Set$.

If we have a "drawable shapes" category with one object $[0, 1]$ and just the identity, this is the trivial category, so $PSh([0, 1]) \cong PSh(\cdot) \cong \text{Set}$ formally, because the point maps to a set, and the natural transformations are functions between sets, satisfying naturality vacuously.

We can intuitively think of one presheaf $F : [0, 1]^{op} \rightarrow \text{Set}$ as mapping the interval to a set of various abstract curves, which is an "abstract drawing". No relationship between these curves in a set is indicated by the formalism. A map between abstract drawings maps curves to curves.

It's not clear to me where this intuition comes from, I hope I'm relating the intuition to the right parts.

To get more sophisticated "abstract drawings", we need to add maps to our category of one object $[0, 1]$. To draw a circle we need to be able to glue the ends of our segments together. So we add $c_0 : [0, 1] \rightarrow [0, 1]$ and $c_1 : [0, 1] \rightarrow [0, 1]$ both defined so $c_x(t) = x$.

Now a functor $F : [0, 1]^{op} \rightarrow \text{Set}$ like before maps to a set of curves, but now one such set could have two curves $\alpha$ and $\beta$ acting as two halves of a circle. They represent this abstractly by requiring $\alpha \circ c_x = \beta \circ c_x$.

Now a natural transformation between such functors must map circles to circles. Because it preserves the structure given by $c_x$.

I hope I wrote that out correctly. A lot of how I understood/interpreted that was via "pattern matching". In particular, I'm not sure why these presheafs can be visualized as making sets of curves from $[0, 1]$.

Mike Shulman (Apr 20 2025 at 05:59):

That seems about right, except that I wouldn't write [0,1] for the category, rather [0,1] is the unique object in that category.

Mike Shulman (Apr 20 2025 at 06:01):

Alex Kreitzberg said:

I'm not sure why these presheafs can be visualized as making sets of curves from $[0, 1]$.

Maybe it would help to think about the case of representable presheaves. If you have an object $X$ of some category that contains [0,1], like topological spaces, then there is a presheaf $Hom(-,X)$ that sends the object [0,1] to the set of maps $[0,1]\to X$ in that category. So in that case, the elements of the presheaf certainly are "curves". The point is to generalize that intuition so that for an arbitrary presheaf you think of the elements as "curves", but in some abstract sense rather than in some concrete canvas $X$.

Alex Kreitzberg (Apr 20 2025 at 06:47):

Oooooh representable presheaves don't form a topos, but presheaves do. So it's fortunate we can have a common intuition for all presheaves.

I'll sit on this (and circle back to over categories)

Simon Burton (Apr 20 2025 at 13:40):

Thankyou @Alex Kreitzberg for asking these questions, it's all very interesting... I'm always afraid to ask stuff like this here because i'm never quite sure what my question is, and the answer tends to be "all of category theory".

One thing i'd like to mention is how the Hom's inherit (algebraic?) structure from the target, so for example, the space of paintings Hom($X$,[0,1]) inherits algebraic operations from [0,1]. You can for example take the minimum of two paintings (or maximum) and this gives a new painting. You can also multipy two paintings.

There's a big theme in mathematics about when/if/how you can recover the points (drawings) of $X$ from the paintings on $X$, such as Hom($X$,[0,1]), or Hom($X$,$\mathbb{R}$) etc.

Alex Kreitzberg (Apr 20 2025 at 14:10):

I appreciate you saying that, I'll end up thinking of one of these questions and spend weeks trying find the right way to phrase it.

I felt like I was only doing okay, so it's a relief to know somebody else finds my questions useful. :laughter_tears:

Alex Kreitzberg (Apr 20 2025 at 14:15):

The inherited algebraic structure on the $\text{hom}(X, [0, 1])$ you described looks like blending modes!

Alex Kreitzberg (Apr 20 2025 at 14:31):

Actually I want to add a point to the "blending mode" observation. Those operations also work for "digital brushes", which fake paint mixing on the canvas.

But even with real art material - your choice between dry pigments, transparent paint, or opaque paint, determines how "additive - average" vs "subtractive" the color mixing behaves. The terminology "burn" and "dodge" came from analog photography, etc. So even with real material you have a concept of "blending mode". You don't need computers for these ideas to be useful.

Alex Kreitzberg (Apr 20 2025 at 15:49):

In light of Simon's comments, I guess the following thought belongs here as well

Here's a trick using "additive blending"

Suppose I have a drawing $\ell : [0, 1] \rightarrow R^2$, and a "brush" $b : R^2 \rightarrow [0, 1]$, like $b(x, y) = e^{-x^2 + -y^2}$ for example. Then I can get a painting $h(x, y) = \int_0^1 b((x,y) - \ell(t))dt$, by "drawing" $\ell$ with the "brush" $b$.

I want to write some product notation analogous to $\langle \ell | b \rangle$ for this (but bracket returns a scalar not a function). I'm not sure what it should be assuming it already exists.

Alex Kreitzberg (Apr 20 2025 at 16:14):

In any case, one question I've been lightly thinking about is whether you can factor out $\ell$ and $b$ from a painting. Or think about a painting as a decomposition $\sum_i c_i``\langle \ell_i | b_i \rangle"$, etc.

Alex Kreitzberg (Apr 20 2025 at 16:40):

I guess simply integrating over a path is the more important basic concept here though, if I think about this in a physics motivated way:

Then the notation is just
$\int_\ell f$ which is fun because of stokes theorem $\int_{\ell} df = \int_{\partial \ell} f$, so maybe there's some cute way to relate this stuff.

Anyways, I guess I'm half rambling about what this stuff makes me think about. Things like the signatures

$R \rightarrow X \times X \rightarrow R \rightarrow R$,

or

$[0, 1] \rightarrow X \times X \rightarrow [0, 1] \rightarrow X \rightarrow [0, 1]$

etc.

David Egolf (Apr 20 2025 at 16:53):

On the topic of "curves" without an explicit canvas - that reminds me of an approach towards imaging reconstruction I explored some years ago. The context was this: we were trying to make an image of some "dots", and aiming to correctly locate the dots even when some of them were very very close to one another.

It ended up being beneficial in multiple ways to represent an image in terms of a list of dot locations (and "brightnesses"), instead of using a function that assigned a brightness value to each point in some discrete gridding of the area to be imaged.

David Egolf (Apr 20 2025 at 16:55):

Or maybe this was switching to a drawing instead of a painting? I'd have to think about this more carefully.

Anyways, thought it might be a fun example to mention.

Alex Kreitzberg (Apr 20 2025 at 17:05):

Yeah! $R^2 \rightarrow [0, 1]$ is different than $Z^2 \rightarrow [0, 1]$, is different than $(R^2 \times [0, 1])^*$ (where the $*$ indicates we want a free monoid)

But they all involve $[0, 1]$ in a way that makes it feel like an "output". Which makes these all feel like they're related to presheafs somehow, imo.

Simon Burton (Apr 20 2025 at 17:11):

@David Egolf aha, this reminds me of the distinction between rasterized graphics (bitmaps) versus vector graphics! There's always interesting reasons to choose one over the other...

John Baez (Apr 20 2025 at 18:02):

Alex Kreitzberg said:

A glance at the nLab articles indicates that apparently you can do a ton more with over categories than with under categories.

That's not exactly right, because every under category is the opposite of some over category, and vice versa.

Puzzle. Why?

However, the answer to the puzzle explains a bit about why over categories are commonly favored.

John Baez (Apr 20 2025 at 18:48):

@Mike Shulman got me to fix the wording of my puzzle.

Alex Kreitzberg (Apr 20 2025 at 20:26):

Solution?

The over category $C / c$ has various properties depending on the properties of $C$.

If $C$ is a topos, then $C / c$ is a topos. If $C$ is a Grothendeick topos, then $C / c$ is a Grothendeick topos. The slice of presheaves is a presheaves on a slice.

Apparently the slice topos over a monoid object is a monoidal topos (which I suppose is where some of the blending modes above come from)!

These properties aren't preserved when constructing under categories.

So one way to answer this question is to note, that for whatever reason, topoi and presheaves are more important than their duals. In particular Set is incredibly important, and a topos, whereas the nlab article [[cotopos]] currently has recorded a single much less important example.

This feels like playing the "why?" game though. It's not entirely clear to me why a cotopos would necessarily be less important, so maybe I'm missing a deeper reason.

(A possibly related exchange question/answer for this: https://math.stackexchange.com/questions/3168833/why-are-presheaves-more-important-than-copresheaves)

John Baez (Apr 20 2025 at 21:05):

I wouldn't say a cotopos is less important than a topos; in algebraic geometry toposes let us study geometry while cotoposes let us study commutative algebra: these are dual viewpoints on the same subject. It's just that every theorem about toposes gives a theorem about cotoposes, and vice versa, in a very straightfoward way. So we don't need a big fat book about cotopos theory to put next to our big fat book about topos theory, containing dualized versions of all the theorems.

Alex Kreitzberg (Apr 20 2025 at 21:19):

So under categories should be really common in commutative algebraic settings.

Then over categories might be more popular because Set is more popular/a common foundation (but not more important)?

Or am I still missing the point?

Alex Kreitzberg (Apr 20 2025 at 21:26):

I guess over categories are probably more popular in Set, but you're allowed to make under categories in Set as well, they just aren't as nice.

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

Whether over categories are more popular, in general, isn't as important or interesting mathematically.

Alex Kreitzberg (Apr 20 2025 at 21:46):

But $\text{hom}$ is a functor into Set, so Set does seem especially important in category theory.

Alex Kreitzberg (Apr 20 2025 at 21:57):

The yoneda embedding lets you realize a category $C$ as a full subcategory inside its category of presheaves, which is a topos.

Alex Kreitzberg (Apr 20 2025 at 22:01):

The contravariant yoneda embedding is just the same operation over $C^{op}$ which still gives a topos.

So in the way we usually do category theory, isn't it easier to get a topos?

John Baez (Apr 20 2025 at 22:31):

Alex Kreitzberg said:

So under categories should be really common in commutative algebraic settings.

Yes. But when algebraic geometers think about commutative algebra, they often work with the opposite of the category $\mathsf{CommAlg}$ of commutative algebras, which they call the category of "affine schemes", $\mathsf{AffSch}$. Then they focus on over categories!

But an over category of an affine scheme is just the opposite of the under category of a commutative algebra. So it's just a matter of preference which you study. Do you like thinking geometrically, or algebraically? Grothendieck's works are largely phrased in terms of schemes and over categories, and this viewpoint led him to invent topos theory as a generalization of set theory, instead of "cotopos theory" as a generalization of "co-set theory".

You're right, our fondness for set theory pushes us to like toposes more than cotoposes. The opposite of the category of sets is the category of complete atomic boolean algebras, and these are very nice, but less famous than sets.

Alex Kreitzberg (Apr 20 2025 at 23:00):

Is there an opposite version of category theory that uses complete atomic boolean algebras for arrowsᵒᵖ instead of sets?

John Baez (Apr 20 2025 at 23:14):

Are you talking about cocategories?

Alex Kreitzberg (Apr 20 2025 at 23:35):

I don't know :laughing:. I'm imagining a mirrored universe and history of mathematics done with only incidental uses of $\text{Set}$, but heavy usage of $\text{Set}^{op}$.

So if in this mirror universe they discover cocategories first, then I guess so!

Anyways this is incidental, I'll circle back to thinking about over categories (and under categories).

John Baez (Apr 20 2025 at 23:41):

Anyway, a cocategory is a category object in $\mathsf{Set}^{\text{op}}$, so if you take the definition of category where you've got a set of objects and a set of morphisms and replace $\mathsf{Set}$ by $\mathsf{Set}^{\text{op}}$, you get the definition of cocategory.

Alex Kreitzberg (Apr 21 2025 at 00:05):

You know, I think I subconsciously interpreted an internal category as dependent on the definition of a category, which I'm understanding as depending on Set.

I'm struggling to imagine what it's like to do mathematics/category theory without sets.

Like $\text{Set}^{op}$ has sets of arrows for homsets in my intuition. It's a member of $Cat$. For this question I was trying to imagine how it could be interpreted as a cocategory belonging to the 2-cocategory $CoCat$, with no reference to Set and therefore Categories.

If a category internal to $\text{Set}^{op}$ automatically does this, or allows this, that's super cool.

(I suspect I'm confusing myself with category theory's primordial ooze)

Mike Shulman (Apr 21 2025 at 00:05):

You can't really do mathematics without sets or some set-like things.

Alex Kreitzberg (Apr 21 2025 at 00:07):

That makes sense to me! Is it possible to explain why that is?

Alex Kreitzberg (Apr 21 2025 at 00:10):

That's something I've just taken for granted, but it feels equally weird for that to be true, and for $\text{Set}^{op}$ to be less commonly used.

Mike Shulman (Apr 21 2025 at 00:10):

Well, I wouldn't go so far as to say there's no possible mathematics that can be done without sets. I mean, simple arithmetic doesn't really use any sets. And maybe some aliens somewhere have invented a kind of higher mathematics that doesn't use sets or any set-like things. But it's an empirical observation that most advanced mathematics that humans have invented uses sets. Probably it's intrinsic to the way we think, organizing things by categorizing and grouping them together.

John Baez (Apr 21 2025 at 00:14):

Alex Kreitzberg said:

That's something I've just taken for granted, but it feels equally weird for that to be true, and for $\text{Set}^{op}$ to be less commonly used.

I made this point before but I'll do it again: working in the opposite of a category is just another way of thinking about working in that category. We don't need a big fat book about $\mathsf{Set}^{\text{op}}$ because there's a 1-1 correspondence between theorems about $\mathsf{Set}^{\text{op}}$ and theorems about $\mathsf{Set}$. For example the theorem "$\mathsf{Set}^{\text{op}}$ has products" is another way of saying "$\mathsf{Set}$ has products". So it's not like we're missing out on anything by talking a lot more about $\mathsf{Set}$ than $\mathsf{Set}^{\text{op}}$.

Mike Shulman (Apr 21 2025 at 00:16):

That's true formally, but I think Alex still has a point. When we introduce a structure in mathematics, we generally define it in terms of sets and functions, not op-sets and op-functions, and I don't think that's just an accident of history.

John Baez (Apr 21 2025 at 00:18):

I agree. People like to think about bunches of things.

John Baez (Apr 21 2025 at 00:20):

Objects in $\mathsf{Set}^{\text{op}}$ are "bunches of predicates" (my rough term for complete atomic boolean algebras), and people also like to think about bunches of predicates - in fact we use them to talk about bunches of things!

Mike Shulman (Apr 21 2025 at 00:21):

But, of course, a CABA is a set equipped with various operations on it.

John Baez (Apr 21 2025 at 00:22):

You're making me want to redo all of mathematics without sets, just to win this argument, but I don't have time.

Mike Shulman (Apr 21 2025 at 00:22):

Haha!

Mike Shulman (Apr 21 2025 at 00:22):

Are we having an argument?

John Baez (Apr 21 2025 at 00:23):

It felt like it, just then.

John Baez (Apr 21 2025 at 00:26):

I'm not really against the idea that there's some sort of primacy of set theory; I'm more trying to get Alex to see that you can flip any statement about $\mathsf{Set}$ into a statement about $\mathsf{Set}^{\text{op}}$ and that this is an example of the duality between geometry and commutative algebra - an example where the "commutative algebra" is a kind of boolean algebra, which is the algebra we often use for logic.

John Baez (Apr 21 2025 at 00:27):

And it would be nice if he took this idea and somehow used it to develop more ideas about the duality between what he was calling "drawings" and "paintings", or what I call "particles" and "fields".

Oscar Cunningham (Apr 21 2025 at 06:17):

(Tangentially related: you can use groups instead of sets as the foundation of mathematics.)

Kevin Carlson (Apr 21 2025 at 17:39):

Well, assuming you believe that you haven't used the concept of "set" at all in describing formal theories clearly enough to try to establish ETCS (ETCG) as a foundation in the first place--that is, you can use groups rather than sets as a foundation relative to a formalization of first-order logic that somehow manages to exist outside the foundation you're in the middle of establishing, which is not obviously what it means to give a foundation of mathematics.

Kevin Carlson (Apr 21 2025 at 17:39):

(But that's a tangent to a tangent, so it should probably spin out to a new thread if people want to engage with it.)