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Stream: community: general

Topic: Looking for popular accounts of Yoneda

Gabriel Goren Roig (Apr 29 2023 at 17:14):

Hi everyone, I'm looking for "popular science" articles (or something similar) that talk about the Yoneda embedding. I naively assumed that there would be plenty, since it is the only meaningful categorical idea that I know how to convey to people with no math background at all (and in my experience it does tend to sound interesting to those people).

However I haven't been able to find much along these lines. The only thing that comes to mind now is Eugenia Cheng's The Joy of Abstraction, an amazing book that I have only started reading. The first chapter seems to be very much in line with what I was looking for, but it doesn't tackle the Yoneda embedding heads-on, and of course I was looking for something shorter. I would really appreciate any help in finding this kind of resource!

Joe Moeller (Apr 29 2023 at 17:25):

Emily Riehl gave a tutorial in ACT2020 about the Yoneda lemma using the category of matrices as a focal point. I don't think it's precisely what you're asking for, but it's good anyway.
https://www.youtube.com/watch?v=SsgEvrDFJsM&ab_channel=AppliedCategoryTheory

John Baez (Apr 29 2023 at 17:33):

The best semi-pop explanation of the Yoneda embedding I've seen is this:

• Tai Danae-Bradley, The Yoneda perspective.

John Baez (Apr 29 2023 at 17:34):

Quote:

From one side, the sculpture looks like an elephant; from another, it looks like two giraffes. But neither angle gives a full description. To really understand the sculpture, we should view it from all possible vantage points. The same idea holds in category theory: more vantage points give more information. And here's the upshot: the Yoneda lemma implies:‍

all vantage points give all information.

This is the essence of the Yoneda perspective mentioned above, and is one reason why categorically-minded mathematicians place so much emphasis on morphisms, commuting diagrams, universal properties, and the like. (Have you noticed?) It's all about relationships!

Gabriel Goren Roig (Apr 29 2023 at 18:21):

John Baez said:

The best semi-pop explanation of the Yoneda embedding I've seen is this:

• Tai Danae-Bradley, The Yoneda perspective.

Thanks for bringing this up! Earlier today I remembered having read this article (and finding it great), but when I went to look it up I was sort of disappointed that it was more mathematically oriented than I remembered. But now I see that I discarded it too quickly: that passage you quoted is really good! I am writing something along the lines of what I wanted (in Spanish) and I think it might be a good idea to translate/paraphrase Tai-Danae there

Gabriel Goren Roig (Apr 29 2023 at 19:12):

Still, I find it surprising that there doesn't seem to be any text online aimed at e.g. Philosophy students, trying to relate the "Yoneda perspective" to ideas about things being determined by their relationships to other things, or debates about the existence of intrinsic properties, or essences. Things that people have been talking about for millennia

Gabriel Goren Roig (Apr 29 2023 at 19:16):

By "aimed at Philosophy students" I don't mean to say a text that engages with specifically philosophical problems, that would be too technical as well... I mean more like "here, I translate part of what makes this cool to me into a language that philosophically minded people can enjoy and find interesting" (without necessarily engaging with the problems that could arise from trying to do so rigorously)

John Baez (Apr 29 2023 at 19:34):

I hope you write such an account.

Morgan Rogers (he/him) (Apr 30 2023 at 17:11):

I've seen at least one instance of what you're describing @Gabriel Goren Roig, but the author overcomplicated/mystified the content for dramatic effect in a way that I disapprove of, so I won't dig it up.

Verity Scheel (May 01 2023 at 10:23):

When talking about the Yoneda perspective outside of mathematical contexts/to non-mathematicians, I think one important point to emphasize is that the lemma works when we can expect to know everything about the thing we are studying, when there is no “hidden information” that we might care about.

This is a reasonable assumption in mathematics (in fact, it’s sort of the point of mathematics IMO), but less often in the real world. For instance, you probably would say that humans could not be fully characterized by their observable properties – no matter how detailed, comprehensive, or hypothetical your collection of properties/relationships for a human is. Humans _probably_ have some hidden state that’s never going to be externally visible (or is random, etc.).

But this is just a small caveat, since the Yoneda perspective can be very useful when talking about observable parts of the real world and you just have to restrict your attention to that to be strictly in the spirit of the maths. (And perhaps this is just saying that isomorphism and equality of objects are different relations.) I just like to bring it up when talking about the philosophy of math.

Morgan Rogers (he/him) (May 02 2023 at 09:32):

Verity Scheel said:

For instance, you probably would say that humans could not be fully characterized by their observable properties – no matter how detailed, comprehensive, or hypothetical your collection of properties/relationships for a human is. Humans _probably_ have some hidden state that’s never going to be externally visible (or is random, etc.).

Avoiding a debate on reductionism, I'd like to point out that the Yoneda lemma depends on the identity morphism. We're not just talking about "externally visible" states/properties: it also includes any "internal" states which are included in the category. The "everything" in question is "everything that has been abstracted into the categorical model", and Yoneda really does talk about that everything. The question of whether we can successfully model an informal 'everything' in a sufficiently rich categorical model is independent of the Yoneda perspective, but I suppose "be aware of which details are captured in your model/the assumptions your model depends on" is always a healthy thing to keep in mind.

Mike Shulman (May 02 2023 at 16:30):

Morgan Rogers (he/him) said:

I'd like to point out that the Yoneda lemma depends on the identity morphism. We're not just talking about "externally visible" states/properties: it also includes any "internal" states which are included in the category.

A philosophical counterpart of this observation is that the principle of "identity of indiscernibles" becomes trivially true when the properties to which you can apply indiscernibility include haecceities (the property of being equal to X).

Evan Washington (May 02 2023 at 21:06):

I can't quite help with a 'pop' account, but I can point to a philosophical account. Patrick Walsh writes about the justification of path induction in homotopy type theory, in response to a pair of papers by Ladyman & Presnell; Walsh ties an account of Yoneda to the view of inferentialism, which is roughly the view that the meaning of expressions is given by how they are used (in inferences, assertions, and so on), and draws lessons about the admissibility of "tonk"-like logical connectives.

John Baez (May 02 2023 at 22:12):

""tonk"-like"?

Verity Scheel (May 02 2023 at 22:31):

Mike Shulman said:

A philosophical counterpart of this observation is that the principle of "identity of indiscernibles" becomes trivially true when the properties to which you can apply indiscernibility include haecceities (the property of being equal to X).

Yeah, this is the better setting for what I said, sorry. If all of your properties are external (e.g. behavioral facts about particular beings in some system), implying that you don't have haecceities, then you don't have a setting to talk about internal properties of them, like identity of beings outside of their behavior. This is not surprising if you make this precise in a formal framework, but I think it's worth pointing out to non-mathematicians. Since I think if you phrase things like the Yoneda principle or identity of indiscernibles non-mathematically, the first intuition is skepticism, “why does a bunch of relationships/properties determine all you need to know about a thing?” (if your intuition for “thing” is not “mathematical object”). And then I get curious about what that says about the world mathematics versus the real world. Anyways, IANAPhilosopher.

Naso (May 03 2023 at 00:01):

John Baez said:

Quote:

From one side, the sculpture looks like an elephant; from another, it looks like two giraffes. But neither angle gives a full description. To really understand the sculpture, we should view it from all possible vantage points. The same idea holds in category theory: more vantage points give more information. And here's the upshot: the Yoneda lemma implies:‍

all vantage points give all information.

This is the essence of the Yoneda perspective mentioned above, and is one reason why categorically-minded mathematicians place so much emphasis on morphisms, commuting diagrams, universal properties, and the like. (Have you noticed?) It's all about relationships!

I was recently reading a book "Seeing That Frees: Meditations on Emptiness and Dependent Arising" where the author Rob Burbea gives a basically identical metaphor for the concept of emptiness in buddhism.
Has anyone applied category theory to buddhism yet? :grinning_face_with_smiling_eyes:

By the way, Carlo Rovelli in his book "Helgoland" also underlined the link between his relational interpretation of quantum mechanics (RQM) and emptiness. I don't know if Rovelli also applies category theory in RQM? I don't recall it mentioned in the book, even though the link to Yoneda seems obvious, at least on a superficial level...

John Baez (May 03 2023 at 00:04):

I know Carlo Rovelli, so I know he doesn't know category theory and is probably not going to learn it in order to express his thoughts in that language. Louis Crane has interesting ideas on relational physics and (n-)category theory. For example

• Louis Crane, Clock and category: is quantum gravity algebraic?

influenced me a lot.

John Baez (May 03 2023 at 00:06):

Some principles stated in this paper:

1. No observation is possible without an observer. Hence there is no Hilbert space associated with a closed universe. Any observer is part of a universe, hence occupies a 3-manifold with boundary, and makes observations on another such with a shared boundary.

2. There is no observation at a distance. Thus the Hilbert spaces in the theory reflect the interface between observer and system. This means they are associated to surfaces.

Evan Washington (May 03 2023 at 02:27):

John Baez said:

""tonk"-like"?

'Tonk' is the name of a logical connective meant to challenge naive notions of proof-theoretic semantics. It's described by its inference rules in a natural deduction system: $\frac{A}{A \mathsf{ tonk } B}$, $\frac{B}{A \mathsf{ tonk } B}$ and $\frac{A \mathsf{ tonk } B}{A}$, $\frac{A \mathsf{ tonk } B}{B}$. If you add these $\mathsf{tonk}$ rules, the proof system becomes trivial, you can prove any $A$. (By "tonk-like," I just meant putative connectives like this.) The lesson we draw is: it's not just the existence of any old inference rules that gives meaningful logical connectives, they have to be the right sort of inference rules. They have to be appropriately "harmonious" with the rest of the inference rules (perhaps part of some 'interlocking adjoints' as Lawvere might put it).

John Baez (May 03 2023 at 03:13):

Thanks! So "tonk-like rules" is a philosopher's idea, a bit like "grue", designed to test our understanding?

Eduardo Ochs (May 03 2023 at 04:27):

Hi @Gabriel Goren Roig,

this is not what you asked for, but I think it's worth mentioning anyway - apologies for the shameless plug, etc, etc.

I spent decades wondering what made some people feel that the Yoneda Lemma and the Yoneda Embedding were "simple" in some sense. It turned out that what I was lacking was a way to draw the Lemma - and the Embedding - in a certain shape that made both the main bijections and the gory details all visible at the same time. That shape is explained in the sections 7.3, 8.4 and 8.5 of this paper,

http://anggtwu.net/math-b.html#2022-md
http://anggtwu.net/LATEX/2022on-the-missing.pdf

Here's an attempt to put Emily Riehl's example from the ACT2020 in my shape:

http://anggtwu.net/LATEX/2020riehl.pdf

Eduardo Ochs (May 03 2023 at 04:28):

And here's the idea of "universal element" in that shape:
universal.png

El Mehdi Cherradi (May 03 2023 at 17:44):

Not sure this relevant to your question, but, in type theory, the Yoneda embedding/lemma is (somewhat exactly) functional extensionality of $\Pi$-types.

Gabriel Goren Roig (May 09 2023 at 18:39):

I appreciate everyone's comments on the topic!

@Evan Washington Thanks for the reference! The relationship with inferentialism seems pretty interesting. There is a straightforward analogy between saying "An object is determined by the arrows into/out of it" versus "The meaning of logical connectives is given by its introduction/elimination rules" which I hadn't thought about at all.

Gabriel Goren Roig (May 09 2023 at 18:39):

@Eduardo Ochs Thank you for your comment. I was aware of your work in this style of diagrams, probably from having seen some previous message of yours in this forum previously. Your notation seems really useful, I hope I get around to studying it a little bit at some point.

Gabriel Goren Roig (May 09 2023 at 18:40):

@Naso Buddhism's emptiness is the main example I had in mind, I guess, of the fact that this topic of existence of intrinsic properties or essences has been discussed for millennia! So I think it's definitely relevant here, but of course, for most people I know, explaining the Yoneda perspective by saying "oh, you know, it's just dependent origination of all phenomena" wouldn't be too... Enlightening :stuck_out_tongue_wink: (sorry, couldn't resist the pun)

Gabriel Goren Roig (May 09 2023 at 18:41):

@Verity Scheel and @Morgan Rogers (he/him) I agree with Morgan in that if you think of categories as models of a certain kind, then the Yoneda perspective is saying that in this kind of models, all properties captured by the models are relational... So I don't think you need to expect to know everything about the thing you are studying. It's just a model, and when working whithin the model, you assume (tacitly) that it's true and it captures everything.

Actually, the fact that you can make mathematical models in which all properties are relational (as witnessed by category theory) can be thought of as evidence for the applicability of these purely relational models. Perhaps it inspires us to think relationally about more things, expecting properties to become relational once we find the right theoretical framework ("the right ambient category") for them. @Verity Scheel you say that the initial intuition for many people about this this sort of relationalism is skepticism. Well, I'm thinking of the Yoneda perspective as a way of challenging that intuition, perhaps not the best one in itself when taken separately, but one that adds to many other ways of motivating or advocating for relational ontology

Gabriel Goren Roig (May 09 2023 at 18:43):

Mike Shulman said:

Morgan Rogers (he/him) said:

I'd like to point out that the Yoneda lemma depends on the identity morphism. We're not just talking about "externally visible" states/properties: it also includes any "internal" states which are included in the category.

A philosophical counterpart of this observation is that the principle of "identity of indiscernibles" becomes trivially true when the properties to which you can apply indiscernibility include haecceities (the property of being equal to X).

I don't quite understand how this relates to Morgan's statement that the Yoneda lemma depends on the identity morphism. Could you please elaborate?

Gabriel Goren Roig (May 09 2023 at 18:45):

John Baez said:

I hope you write such an account.

By the way I did write a first draft for this. It's in Spanish for the time being and I'm waiting to receive and incorporate some feedback on it, but I have in mind a translation to English as well if people seem to like it

John Baez (May 09 2023 at 19:18):

Great!

By the way, I think it would be hard, but interesting, to try to define what you mean by "mathematical models in which all properties are relational" to the level of precision where I could take two models (I think mathematicians call them "theories") and prove that all properties are relational in one, but not in another.

John Baez (May 09 2023 at 19:20):

If I had to do this, I would try by saying that "all properties are relational" in a theory iff there's a certain large amount of symmetry in the theory.

John Baez (May 09 2023 at 19:23):

Physicists have been thinking about relationalism and symmetry at least since Galileo and Leibniz. In 1876 Maxwell wrote:

Our whole progress up to this point may be described as a gradual development of the doctrine of relativity of all physical phenomena. Position we must evidently acknowledge to be relative, for we cannot describe the position of a body in any terms which do not express relation… There are no landmarks in space; one portion of space is exactly like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, sounding, wind or tide, and we cannot tell in which direction we are going. We have no log which we can case out to take a dead reckoning by; we may compute our rate of motion with respect to the neighboring bodies, but we do not know how these bodies may be moving in space.

John Baez (May 09 2023 at 19:24):

3 years before Einstein was born!

John Baez (May 09 2023 at 19:26):

And the connection between symmetry and the Yoneda embedding theorem is very interesting. For example, this theorem has as a special case Cayley's theorem, which could be called the fundamental theorem of group theory. (It says every group can be seen as a subgroup of the group of permutations of some set.)

Simon Willerton (May 09 2023 at 20:58):

I wouldn't say that Cayley's Theorem is a special case, but rather that it's a consequence of a special case. (Unless I'm thinking of a different special case!)

Let $G^\mathrm{reg}$ be $G$ equipped with the right-regular action of $G$ on it, ie. $g\in G$ acts by multiplication on the right. So $G^\mathrm{reg}$ is an object in the category of right $G$-sets (sets equipped with an action of $G$ on the right).

The special case of the Yoneda Embedding Theorem says that $G$ is isomorphic to the group of automorphisms (in the category of right $G$-sets) of $G^\mathrm{reg}$. Looking at the forgetful functor from right $G$-sets to sets then will give you Cayley's Theorem.

John Baez (May 09 2023 at 22:26):

I was thinking of this: Think $G$ as a one-object category. Apply the Yoneda embedding theorem to this category. This embeds $G$ in a full and faithful way into the category of $G$-sets. When we look at the image of this embedding, we get a category equivalent (and in fact isomorphic) to $G$.

What is the image of the embedding? It has one object, the $G$-set you're calling $G^{\text{reg}}$, and a bunch of isomorphisms $G^{\text{reg}} \to G^{\text{reg}}$ as morphisms. So we're seeing that $G$ is isomorphic to some subgroup of the group of all permutations of the set $G$.

John Baez (May 09 2023 at 22:30):

I think I'm saying the same thing as you, actually, but to me this is feels so much like "just a special case" of the Yoneda embedding theorem that I don't think of it as a "consequence".

Simon Willerton (May 10 2023 at 08:25):

The main reason I mention it is that I've seen confusion caused by people having this as a slogan - "Yoneda embedding for a group is Cayley's Theorem". For instance, thinking that it involves a functor from $G$ to sets rather than to $G$-sets.

Also, the Yoneda embedding result is even cooler than Cayley's Theorem: the only permutations of a group that commute with the right action are those coming from left multiplication by a group element!

John Baez (May 10 2023 at 15:35):

Well, the unique representable presheaf on G is a functor from G to Set.

John Baez (May 10 2023 at 15:40):

But the Yoneda embedding gives a functor from G to G-sets.

John Baez (May 10 2023 at 15:43):

I certainly agree that the common statement of Cayley's Theorem as "every group is isomorphic to a subgroup of some permutation group" is pathetically weak, practically just like patting someone on the back and reassuring them that somewhere out there that permutation group exists - what good is that?

Simon Willerton (May 10 2023 at 15:49):

Slander! :-) I didn't say that Cayley's Theorem is pathetically weak.

Anyway, I thought that the good in Cayley's Theorem was along the following lines. People studied symmetries of things and studied groups of such symmetries. Then someone axiomatized what a 'group' is without any reference to symmetry, then Cayley's Theorem says, yes we haven't done anything stupid and added anything weird in our axiomatization, we are still essentially looking at groups of symmetries.

John Baez (May 10 2023 at 15:51):

But that's sort of a "pat on the back" result, something that makes you feel good. I guess sometime I should look around and study more carefully how people actually use Cayley's Theorem to prove other results... not just sleep better at night.

John Baez (May 10 2023 at 15:53):

And that leads me back to the main topic of this thread. There's the "feel-good" aspect of the Yoneda embedding theorem: "it's really cool, dude - everything is relational! :star_struck: " And then there's the various ways people actually use the Yoneda embedding theorem to prove stuff. And there seems to be a gap between these two. It would be good to study that gap, and maybe bridge it.

Simon Willerton (May 10 2023 at 15:57):

And my point was that "The Yoneda embedding for a group is just Cayley's Theorem" is not a good slogan. (I've added a word there.) The result about the automorphisms of the regular action is perhaps more useful. (Can you use it to show something like the group of invertible natural transformations of the identity functor on the category of $G$-sets is the centre of $G$?)

John Baez (May 10 2023 at 16:02):

I think the just is the kind of thing I'd say to get people over their fear of the Yoneda embedding theorem. So you're saying that once they're over their fear, I should admit I was lying.

Simon Willerton (May 10 2023 at 16:05):

I think that you tell people you're lying when you're lying and tell them you'll tell them the truth when they have lost their fear! (I'm not sure that would work as well though.)

John Baez (May 10 2023 at 16:06):

Emily Riehl gives a simple example of how you can actually use Yoneda - it's a proof that multiplication distributes over addition:

.

Anyone seriously pondering popular accounts of Yoneda should ask themselves: how is the "everything is relational" pop summary of Yoneda connected to this? Can you argue that multiplication distributes over addition because everything is relational???

John Baez (May 10 2023 at 16:09):

(More realistically, Yoneda is just one step of this argument: the universal properties of addition and multiplication are crucial.)

Simon Willerton (May 10 2023 at 16:12):

The fact that Set is Cartesian closed also plays an important role.

John Baez (May 10 2023 at 16:14):

Yes. So, the only thing Yoneda does here is show you that to prove $A \times (B + C) \cong A \times B + A \times C$, it's enough to compare maps out of the sets on each side of this "equation".

John Baez (May 10 2023 at 16:15):

That's the "everything is relational" part.

John Baez (May 10 2023 at 16:16):

But then we have to use that $\times$ and $+$ can be understood "relationally": i.e. we can completely understand a product or coproduct in terms of the maps out of it (in the cartesian closed case).

So the idea that "everything is relational - everything can be completely understood by the maps out of it (or into it)" is not just Yoneda here; it's the whole idea of defining things by universal properties.

Simon Willerton (May 10 2023 at 16:17):

I often dislike it when the Yoneda embedding is called in to prove things. Maybe because I'm a bit constructivist. There are occasions where people show $Hom(A, C) \cong Hom(B, C)$ naturally and then say "Thus by Yoneda $A \cong B$" when, in fact, they could, with not much more work, unpack it all and give an explicit isomorphism $A\cong B$.

John Baez (May 10 2023 at 16:19):

E.g. by taking $C = B$.

Simon Willerton (May 10 2023 at 16:22):

Well in the cases I'm thinking of, the isomorphism between $Hom(A, C)$ and $Hom(B, C)$ is a bit messy like a string of many isomorphisms, so chasing the identity from $Hom(B, B)$ to $Hom(A, B)$ is not trivial.

Simon Willerton (May 10 2023 at 16:25):

See, for example, page 111 of Fausk, Hu and May, http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html

Mike Shulman (May 10 2023 at 16:31):

But if the natural isomorphism really is defined constructively, then by plugging in $1_B$ your proof assistant should be able to $\beta$-reduce it for you and produce the map $A\to B$. (-:

Simon Willerton (May 10 2023 at 16:33):

In this case, proof assistant was PhD student. :smile:

Mike Shulman (May 10 2023 at 16:34):

With tongue a little further out of my cheek, in my experience it happens fairly frequently that it's easy to give a natural isomorphism $Hom(A,C) \cong Hom(B,C)$, and easy to write down an explicit map $A\to B$, but not so easy to show that the latter explicit map is an isomorphism without giving the natural isomorphism of hom-sets and showing that its underlying map $Hom(B,C) \to Hom(A,C)$ is induced by your map $A\to B$.

Simon Willerton (May 10 2023 at 16:39):

A strategy is to take $C = A$, get the map $B\to A$ and then show (possibly with difficulty!) that it is the inverse you require.

Mike Shulman (May 10 2023 at 17:17):

Well, yes, you can always $\beta$-reduce the Yoneda proof to get one that doesn't use it. But a normalized proof isn't always shorter or easier to come up with.

Reid Barton (May 10 2023 at 17:17):

Or easier to understand

Jencel Panic (Jun 29 2023 at 06:59):

In my mind, trying to explain the Yoneda lemma philosophically, would amount to in a very interesting philosophical text (as it does now, in this discussion), but a lousy explanation of the concept.

Jencel Panic (Jun 29 2023 at 07:03):

Otherwise, the first philosophical slogan that comes to mind when I think of CT is Wittgenstein's "The world is the collection of facts, not of things."

David Michael Roberts (Jun 29 2023 at 10:56):

Jencel said:

Otherwise, the first philosophical slogan that comes to mind when I think of CT is Wittgenstein's "The world is the collection of facts, not of things."

I once thought I could explain a bunch of the early parts of the Tractatus using the Yoneda embedding and internal logic. Sadly, never wrote it up...

Andrius Kulikauskas (Jun 29 2023 at 14:10):

@Jencel what do you think of my presentation "The Yoneda Embedding Expresses Whether, What, How, Why" ? https://www.math4wisdom.com/wiki/Research/YonedaEmbeddingFoursome I very much appreciate your critique.

@David Michael Roberts I am curious to learn more about your ideas.

Jencel Panic (Jun 29 2023 at 19:02):

@David Michael Roberts Really, me too! There is certainly something to it, as Wittgenstein was mentored by Russell, who is a definitely a pioneer in CT (although not many people give him credit for that)

Jencel Panic (Jun 29 2023 at 19:04):

We should open a topic about this in #philosophy maybe. I can contribute something as well.

Jencel Panic (Jun 29 2023 at 19:04):

@Andrius Kulikauskas I will check it!

Leopold Schlicht (Jun 29 2023 at 21:17):

David Michael Roberts said:

Jencel said:

Otherwise, the first philosophical slogan that comes to mind when I think of CT is Wittgenstein's "The world is the collection of facts, not of things."

I once thought I could explain a bunch of the early parts of the Tractatus using the Yoneda embedding and internal logic. Sadly, never wrote it up...

Wow, that sounds fascinating!

David Michael Roberts (Jun 30 2023 at 01:58):

@Andrius Kulikauskas @Jencel the best I can do is look at my comments written in the margin of my copy of Tractatus, and relay them.

David Michael Roberts (Jun 30 2023 at 07:11):

So, for instance, I wrote "Yoneda" next to

2.0123 If I know an object I also know all its possible occurrences in states of affairs. (Every one of these possibilities must be part of the nature of the object.) A new possibility cannot be discovered later.

Andrius Kulikauskas (Jul 09 2023 at 13:04):

@David Michael Roberts Thank you! Yes, this is of interest to me and certainly also @Kirby Urner Please do share more! It is interesting the different forms this idea may take for Wittgenstein. I could go from there and see where other aspects of the Yoneda embedding come in such as the do-nothing action (the identity element).

David Michael Roberts (Jul 10 2023 at 01:27):

I can share with you a complaint I have, about an early entry (1.21) which I think is not quite coherent

https://philosophy.stackexchange.com/questions/36140/what-is-wittgenstein-claiming-when-he-says-that-each-thing-can-be-the-case-or-n

I'm not sure I'm satisfied with the answer

Posina Venkata Rayudu (Jul 10 2023 at 01:36):

Gabriel Goren Roig said:

Hi everyone, I'm looking for "popular science" articles (or something similar) that talk about the Yoneda embedding. I naively assumed that there would be plenty, since it is the only meaningful categorical idea that I know how to convey to people with no math background at all (and in my experience it does tend to sound interesting to those people).

However I haven't been able to find much along these lines. The only thing that comes to mind now is Eugenia Cheng's The Joy of Abstraction, an amazing book that I have only started reading. The first chapter seems to be very much in line with what I was looking for, but it doesn't tackle the Yoneda embedding heads-on, and of course I was looking for something shorter. I would really appreciate any help in finding this kind of resource!

Just in case it fits your bill, I'm sharing a note on Yoneda lemma/embedding I wrote sometime ago (read at your own risk ;)
Yoneda_Lemma.pdf

I must also hasten to share with you the corrections Professor Andree Ehresmann was kind enough to provide:
Yoneda_Embedding.pdf

Here's a whittled-down version:
Yoneda_One_Morphism.pdf

Jencel Panic (Jul 10 2023 at 20:49):

David Michael Roberts said:

So, for instance, I wrote "Yoneda" next to

2.0123 If I know an object I also know all its possible occurrences in states of affairs. (Every one of these possibilities must be part of the nature of the object.) A new possibility cannot be discovered later.

The Yonesa lemma is kind of the opposite, though - If I know the possible occurrences of the object, then I know the object.

Jencel Panic (Jul 10 2023 at 20:58):

David Michael Roberts said:

I can share with you a complaint I have, about an early entry (1.21) which I think is not quite coherent

https://philosophy.stackexchange.com/questions/36140/what-is-wittgenstein-claiming-when-he-says-that-each-thing-can-be-the-case-or-n

I'm not sure I'm satisfied with the answer

I think that you are thinking of propositions, whereas W. more likely refers to (atomic) facts, which play the role of axioms in his system. So this proposition just says that axioms are independent of one another.

Jencel Panic (Jul 10 2023 at 21:58):

Notice the formulation of the first proposition, to which this one is related: "The world is all that is the case" He doesn't say "everything that is true", I think that it is because we can derive infinitely many true propositions from a given set of facts, but those are just our invention, not part of "the world".

Kirby Urner (Jul 11 2023 at 19:47):

@Andrius Kulikauskas et al, I've been delving into this "popular account":
https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-14/

Re Wittgenstein: I've been much more a scholar of his later work but then LW always point back to his Tractatus as somewhat complementary. Invoking his "meaning from usage" (but also "aspect shifts" i.e. gestalts) seems consistent with the doctrine of "emptiness" in Zen etc. i.e. meaning is not from "pointing" to some essentially private self object and that goes for the word "I" too.

Kyle Wilkinson (Jul 14 2023 at 13:29):

Here is a presentation on a neuroscience topic by Dr. Alexander Maier which discusses the Yoneda lemma. It is an informal account.
I know almost nothing about neuroscience, but the claim is that enriched categories and the Yoneda lemma are applied to help with a paradox in color perception known as the inverted spectrum hypothesis. If the claim is true, that certainly is interesting.

https://www.youtube.com/watch?v=4GJ4UQZvCNM

The referenced paper: https://www.sciencedirect.com/science/article/abs/pii/S1053810022000514

Matteo Capucci (he/him) (Jul 15 2023 at 11:37):

I recorded my qualms with that (and other) usages of category theory in this post.

Ralph Sarkis (Jul 15 2023 at 12:02):

A comment under that video

Now I know what to say when someone asks me for applications of category theory :laughter_tears:

John Baez (Jul 15 2023 at 12:05):

Wow, this gives "I dropped what I was doing" whole new depths of sleazy connotations.

John Baez (Jul 15 2023 at 12:06):

Let's stop right here.

Kyle Wilkinson (Jul 15 2023 at 14:14):

Matteo Capucci (he/him) said:

I recorded my qualms with that (and other) usages of category theory in this post.

Excellent, thanks for pointing this out! I hope I haven't done harm by posting this, since I am not (yet) able to formally vet things the way many of you are.

Also, is it just me or are there more "squishy" interpretations of category theory concepts floating out there than for other fields of math?

John Baez (Jul 15 2023 at 14:16):

I think there are. Category theory has an instant appeal for people trying to do revolutionary things.

Kyle Wilkinson (Jul 15 2023 at 14:18):

John Baez said:

Wow, this gives "I dropped what I was doing" whole new depths of sleazy connotations.

Yes, I saw this too. I've also noticed there is someone who repeatedly posts about oddball "hyperduality" ideas under most category theory videos.

John Baez (Jul 15 2023 at 14:19):

Ah, the wonderful underworld of YouTube comments... I'm glad I don't allow comments on my YouTube videos.

Jencel Panic (Jul 16 2023 at 10:18):

Matteo Capucci (he/him) said:

I recorded my qualms with that (and other) usages of category theory in this post.

I think the Yoneda lemma has potential to become a second Gödel’s second incompleteness theorem (pun intended haha) in the list of results that are abused for all kinds of purposes by people who are vaguely familiar with them.

Morgan Rogers (he/him) (Jul 17 2023 at 10:04):

Matteo Capucci (he/him) said:

I recorded my qualms with that (and other) usages of category theory in this post.

This paragraph!!!

[...] they start by assuming a specific category of ‘qualia’, or other things, and then they claim to be able to uniquely pin down the objects therein using isomorphism classes of representable presheaves over it. But this is circular: everything is determined by the choice of morphisms they make when defining the category at the start, so they can distinguish objects only insofar as they already assumed they could do so.

Thank you for pinning down the flaw in these supposed applications of Yoneda, very valuable work @Matteo Capucci (he/him)

Morgan Rogers (he/him) (Jul 17 2023 at 10:07):

By the way, there are a couple of typos in the inline math parts of your blog post @Matteo Capucci (he/him) ;)

Todd Trimble (Jul 31 2023 at 02:28):

John Baez said:

But that's sort of a "pat on the back" result, something that makes you feel good. I guess sometime I should look around and study more carefully how people actually use Cayley's Theorem to prove other results... not just sleep better at night.

Here's one I learned about some time back that's sort of cute: given a prime $p$, every finite group embeds into one of the form $GL_n(\mathbb{Z}_p)$ [use Cayley to embed in a symmetric group of permutations, then interpret those as permutation matrices].

What good is that? Well, for one, it gives a nice way of proving the existence of $p$-Sylow subgroups for any finite group. The basic idea is to show that if you can embed a group $G$ into a group known to have a $p$-Sylow subgroup, then $G$ does as well. But $GL_n(\mathbb{Z}_p)$ has a $p$-Sylow subgroup (consisting of "unitriangular matrices", i.e., all 1's down the diagonal, and 0's below the diagonal). See Theorem 3.4 here.

Jean-Baptiste Vienney (Jul 31 2023 at 04:55):

In any symmetric monoidal category, for every permutation $\sigma \in \mathfrak{S}_{n}$, you get a natural transformation $\sigma:A^{\otimes n} \rightarrow A^{\otimes n}$. Now for any group $G$, by Cayley theorem, you can choose a monomorphism $m:G \hookrightarrow \mathfrak{S}_{n}$ for $n = |G|!$ and it gives me a natural transformation $m(g):A^{\otimes n} \rightarrow A^{\otimes n}$ for every $g \in G$. I find it really cool. Groups are an abstract theory of symmetries . Symmetric monoidal category are somehow more practical, they only know how to swap two objects $A \otimes B \rightarrow B \otimes A$ and thus how to permute a list of object $A_{1} \otimes ... \otimes A_{n} \rightarrow A_{\sigma(1)} \otimes ... \otimes A_{\sigma(n)}$ for some permutation $\sigma$. But thanks to Cayley theorem, you can let groups enter into the party in any symmetric monoidal category.

I find this useful for understanding some philosophy but if I want for instance to make $\mathbb{Z}_{n}$ enter into the party, I will not use Cayley theorem but rather interpret $k \in \mathbb{Z}_n$ as the cyclic permutation $c(k)$ of $\mathfrak{S}_n$ which sends $l \in [1,n]$ to the representative of $l + k$ modulo $k$ in $[1,n]$. Now I can look at the cyclic permutations $c(g):A^{\otimes n} \rightarrow A^{\otimes n}$ for every $g \in \mathbb{Z}_n$ and for instance take the coequalizer of all the $c(g)$ which will gives a natural transformation $A^{\otimes n} \rightarrow (A^{\otimes n})_{\mathbb{Z}_n}$ or the equalizer of all the $c(g)$ which will gives a natural transformation $(A^{\otimes n})^{\mathbb{Z}_n} \rightarrow A^{\otimes n}$. If you take the category of modules, you'll get the "cyclic powers" of your modules, either the space of vectors invariant by cyclic permutation with the equalizer or the space of cyclic tensors with the coequalizer. And if you take the category of sets, you'll get either the list invariant by cyclic permutations or the cyclic lists. If your symmetric monoidal category is enriched over the category of abelian groups, then if you replace $c(g)$ by something like $(-1)^{n}c(g)$, you get the modules involved in cyclic (co)homology which are kind of anticommutative cyclic power.

Well, that's a bit far fetched, but that's what evokes me the Cayley theorem: the idea of letting a group act on the tensor powers of an object in a symmetric monoidal category by reminding that groups are just an abstract way to encode permutations and that symmetric monoidal categories know about these latters.