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Stream: learning: questions

Topic: Hegel dialectics

Peiyuan Zhu (Feb 28 2023 at 19:32):

How to understand William Lawvere's comment on adjunction as becoming and Hegel dialectices?

John Baez (Feb 28 2023 at 21:57):

Go to a Zen monastery and meditate for 5 years.

Peiyuan Zhu (Feb 28 2023 at 22:06):

Lol is this real or is it a joke?

John Baez (Feb 28 2023 at 22:06):

Well, you asked how to understand it, and I told you.

Peiyuan Zhu (Feb 28 2023 at 22:08):

Is it really that esoteric? I thought it's something everyone knows given so much online videos and papers about it...

Peiyuan Zhu (Feb 28 2023 at 22:12):

And I thought it has to do with physics too...

John Baez (Feb 28 2023 at 22:12):

Okay - then just watch those videos and read those papers.

Peiyuan Zhu (Feb 28 2023 at 22:14):

May I ask why Zen monastery instead of a Christian church or a Zarathustra temple?

Jean-Baptiste Vienney (Feb 28 2023 at 22:15):

Maybe you can ask on the nForum. Urs Schreiber and David Corfield are interested at the same time by category theory and Hegel. For instance they wrote a massive entry on a logico-categorical interpretation of his book Science of Logic and they also wrote on the philosophical ideas of Lawvere in the entry William Lawvere. They could probably help you. You can ask there: nforum. Moreover David Corfield is an authentic philosopher so maybe he's more able to answer the type of question "how to understand X?".

John Baez (Feb 28 2023 at 22:17):

Peiyuan Zhu said:

May I ask why Zen monastery instead of a Christian church or a Zarathustra temple?

No particular reason. It just sounded funnier.

John Baez (Feb 28 2023 at 22:18):

I find a question of the form "how to understand X?" to be a bit lazy, so I gave a lazy answer - basically, "think about it!"

John Baez (Feb 28 2023 at 22:19):

But watching videos and reading papers is also good.

John Baez (Feb 28 2023 at 22:20):

The nLab is also good.

Peiyuan Zhu (Feb 28 2023 at 22:24):

John Baez said:

I find a question of the form "how to understand X?" to be a bit lazy, so I gave a lazy answer - basically, "think about it!"

To follow the relationalism idea, I believe my understanding of a subject is always in relational to everyone else's understanding of the subject, hence I asked. If it's a very niche topic that only a few people knows my strategy of approach would be very different from a topic that everyone knows. So I think it makes sense to know what everyone else knows.

John Baez (Feb 28 2023 at 22:30):

It would be great if we could know what everyone else knows, but even if they want to, they can't simply dump that knowledge into your brain.

John Baez (Feb 28 2023 at 22:32):

I feel I understand a fair amount about Lawvere's ideas on Hegelian dialectics and adjunctions, but it would take a huge amount of work for me to pass that understanding on to you.

John Baez (Feb 28 2023 at 22:33):

On the other hand if you ask me whether every summand of a free module is projective, this is extremely easy to answer: "yes".

John Baez (Feb 28 2023 at 22:33):

Somewhere in between is a question like "why is every summand of a free module projective?" This is harder to answer, but still infinitely easier than explaining Lawvere's ideas on Hegelian dialectics and adjunctions.

Luckily for you, people have already done a lot of work explaining Lawvere's ideas on Hegelian dialectics and adjunctions. As you said, a lot has been written about this.

Peiyuan Zhu (Feb 28 2023 at 22:35):

John Baez said:

It would be great if we could know what everyone else knows, but even if they want to, they can't simply dump that knowledge into your brain.

But I think it signifies what resources may help to understand it. I also don't think "just thinking" without "first experiencing" would help me understand anything. It's always an interaction with a community of thoughts. I'm not saying thinking isn't important. I'm saying it has to be an activity that involves both thinking and experiencing. If I just start with William Lawvere's papers I might be doomed. I think any idea especially ones that has to do with so many different things (which in this case include physics that I'm not familiar with) that makes it complex is hard to be copied.

Peiyuan Zhu (Feb 28 2023 at 22:36):

I think last time I read into this topic I was stuck at https://ncatlab.org/nlab/show/space+and+quantity

John Baez (Feb 28 2023 at 22:39):

Well, good luck! If you ask a question that's precise and easy to answer, I might answer. If you say "how can I understand X?", where X is some large topic that's famously hard to understand, I won't. It's too much like asking "how can I play the flute?"

I usually try to resist saying anything when you ask these very broad questions. But in this case I could not resist.

Peiyuan Zhu (Feb 28 2023 at 22:40):

I remember someone saying "knowing the question is almost the same as knowing the answer"

John Baez (Feb 28 2023 at 22:41):

Right! And if you don't almost know the answer, you won't understand the answer even if someone answers your question.

Jean-Baptiste Vienney (Feb 28 2023 at 22:42):

I think also people here are not philosophers. I really think that some philosophers like Wittgenstein are really helping me to find ideas for doing math but I'm not so good at writing philosophy so every time I try to explain it, it's so fuzzy that I prefer not trying to do it. I think you seem to have more background in philosophy than in mathematics , so you speak in a way that is disturbing for people here because we are used to another kind of language. I don't think it is bad to ask "how can I understand X?" but that's usually perceived bad by math people

Jean-Baptiste Vienney (Feb 28 2023 at 22:43):

I think it's good to speak in a very imprecise way to get plenty of ideas but people don't always like it.

John Baez (Feb 28 2023 at 22:43):

Yes. I should avoid responding negatively to Peiyuan's questions because we're supposed to encourage questions here. So if I just stay quiet maybe someone will give him a useful answer.

Peiyuan Zhu (Feb 28 2023 at 22:49):

I remember getting quite lost in abstractions without meanings when I was doing pure math at school. And I felt like it was mentally torturing, it was like brainwashing to the extent that it becomes some "a priori structure" in my mind, and I didn't like doing it that way at all. Since I graduated and started work I did some thinking and reading into philosophy, I changed to a different approach that's more application driven and allow imprecise speculations, although the goal still being to lock the dragon into the cage, so of course only if there's a potentially decent rate of convergence towards precision, and a rate of convergence to me means the question is a sensible one.

Peiyuan Zhu (Feb 28 2023 at 22:50):

I think in this case at least the discussion has converged to the Rel category and evolutionary dynamics after a while. But when it first started it's always vague and I tend to think it's unavoidable if I want to do something that's really meaningful to me.

Ellis D. Cooper (Feb 28 2023 at 22:51):

ChatGPT:
ME: What would be a particular concept or object discussed by both of them?
IT: While Lawvere and Hegel come from different fields of study, there are some conceptual overlaps between their work.
One example of a concept or object discussed by both Lawvere and Hegel is the notion of "structure." Lawvere's work in category theory is concerned with the study of mathematical structures and their relationships, while Hegel's philosophy emphasizes the importance of understanding the structures and relationships that underpin social and political phenomena.
Both Lawvere and Hegel are interested in understanding how complex systems emerge from the interactions and relationships between simpler components or elements. For Lawvere, this involves studying the structure and relationships of different mathematical objects and categories, while for Hegel, this involves studying the dialectical relationships between ideas, individuals, and social institutions.

Overall, while Lawvere and Hegel come from different fields of study and have different areas of focus, they share an interest in understanding the underlying structures and relationships that shape complex phenomena, whether in mathematics, philosophy, or society.

John Baez (Feb 28 2023 at 22:52):

:smiling_devil:

Peiyuan Zhu (Feb 28 2023 at 23:03):

John Baez said:

:smiling_devil:

I don't know how many people can relate to that experience of me because I think if I let go the imprecise part math becomes so dull and I will regret it because I will keep seeing deeper structures coming up again and again that changes the very fundamentals, it's like getting hit by a baseball bat again and again until I feel disoriented. But if I begin by thinking of the fundamentals then this process is much more interesting and less torturing. However as I'm not born into a mathematical physicist family so I'm quite far from those people who're actually working on the fundamentals and what they think can be quite distant, and I have a lot to learn in the process. But I think I can understand why John doesn't like vague questions because it's not a simple true / false judgement and the mathematical community somehow has to reinforce this like an immune system because mathematics would then lose what it has achieved so far. This is also why I like reading Jean-Pierre Aubin's book, including his criticism of Nicolas Bourbaki, although initially I found it too hard to understand because it's so different from the math textbooks that I'm used to in school, but later on as I pay attention to all the non-mathematical references in sociology, economics, biology that he used to explain math everything started to make a lot of sense to me.

John Baez (Feb 28 2023 at 23:15):

I have lots of vague imprecise thoughts, and I've spent a lot of time studying philosophy. But this is a category theory forum so I try to talk about math here.

I don't study math unless it has some immediate intuitive meaning to me. So for example when I talk about an invertible object in a symmetric monoidal category I have a kind of mental image of it - and more importantly, lots of ways of relating it to other things that I know about: phases in quantum mechanics, electromagnetism as a gauge theory, line bundles in geometric quantization, Picard varieties in algebraic geometry, Fourier transforms, etc. etc. Abstractions are interesting to me only when they relate to real life.

John Baez (Feb 28 2023 at 23:16):

My main problem is not exactly with "vague" questions, it's with lazy questions: questions that sound like "please dump your carefully developed understanding of X into my open brain".

John Baez (Feb 28 2023 at 23:17):

If it were possible to do such a "brain dump" I would, but it's not.

John Baez (Feb 28 2023 at 23:17):

So I much prefer a question where someone has clearly thought about something already - so that a paragraph can really help them.

Peiyuan Zhu (Feb 28 2023 at 23:19):

John Baez said:

My main problem is not exactly with "vague" questions, it's with lazy questions: questions that sound like "please dump your carefully developed understanding of X into my open brain".

I don't think that initial question mean such a thing. But at least there would be useful resources on how that understanding was developed. I cannot be out of nowhere. I think any question being asked gives a possibility of reducing uncertainty. In this case you're thinking of something will reduce the uncertainty to zero. This will never happen.

John Baez (Feb 28 2023 at 23:20):

I'm not talking about "zero uncertainty". It's impossible to have zero uncertainty when talking about Lawvere's understanding of Hegel - it's famously hard to understand Hegel, and also to understand what Lawvere was saying about Hegel.

John Baez (Feb 28 2023 at 23:21):

But for example if you're asking "why did Lawvere think adjunctions were somehow related to the Hegelian dialectic", to say anything useful I'd have to first know how much you know about adjunctions... and separately, how much you know about the Hegelian dialectic.

John Baez (Feb 28 2023 at 23:22):

If you don't know much about adjunctions, well, that's a good subject for this forum. So you could take a specific adjunction, like the adjunction between topological spaces and sets, and try to work out what it's doing.

Peiyuan Zhu (Feb 28 2023 at 23:22):

What you said here "please dump your carefully developed understanding of X into my open brain" sounds like it -- if the dump can happen then uncertainty would reduce to close to zero. I didn't expect the uncertainty to be reduced to zero or close to zero either. Also I have thought about this before but the physics part made it difficult. It looks like a thermodynamic theory of spacetime.

Peiyuan Zhu (Feb 28 2023 at 23:23):

Is there a simple example of adjunction that I can look into?

John Baez (Feb 28 2023 at 23:23):

I just gave you one, right?

John Baez (Feb 28 2023 at 23:27):

If you understand that one then you might (might!) be able to understand Lawvere's thoughts on [[adjoint modalities]], which are maybe the easiest way to understand his thoughts on the Hegelian dialectic.

John Baez (Feb 28 2023 at 23:27):

But it's not easy.

Peiyuan Zhu (Feb 28 2023 at 23:31):

John Baez said:

I have lots of vague imprecise thoughts, and I've spent a lot of time studying philosophy. But this is a category theory forum so I try to talk about math here.

I don't study math unless it has some immediate intuitive meaning to me. So for example when I talk about an invertible object in a symmetric monoidal category I have a kind of mental image of it - and more importantly, lots of ways of relating it to other things that I know about: phases in quantum mechanics, electromagnetism as a gauge theory, line bundles in geometric quantization, Picard varieties in algebraic geometry, Fourier transforms, etc. etc. Abstractions are interesting to me only when they relate to real life.

Ok so you agree with having some example that has immediate intuitive meaning at hands. In this case can I ask how adjunction come up in physics or computer science? I did saw it appear in thermodynamics https://www.semanticscholar.org/paper/Landauer’s-Principle-as-a-Special-Case-of-Galois-Kycia and programming languages https://en.wikipedia.org/wiki/Abstract_interpretation. Is there simple examples from these two fields?

Peiyuan Zhu (Feb 28 2023 at 23:33):

For instance, the binary search of fixed point as a computer program, is that based on an adjunction?

Peiyuan Zhu (Feb 28 2023 at 23:37):

I'll read into Topoi chapter 15 and see if my questions can be resolved.

John Baez (Feb 28 2023 at 23:39):

In this case can I ask how adjunction come up in physics or computer science?

Adjunctions are absolutely everywhere! But I'll just say this: a lot of theoretical computer science uses 'monads' - they're considered fundamental in some approaches. Every monad is connected to an adjunction (and vice versa).

Peiyuan Zhu (Feb 28 2023 at 23:42):

I see, I do see these concepts coming up quite a lot in this forum.

Peiyuan Zhu (Feb 28 2023 at 23:42):

I can also see "approximate inverse" come up quite a lot everywhere.

David Egolf (Feb 28 2023 at 23:42):

Monads and adjunctions are things that sound super important and interesting to me, but still scare me! They are on my "hopefully actually learn soon" mental list. :upside_down:

John Baez (Feb 28 2023 at 23:44):

Adjunctions are incredibly wonderful - category theory without adjunctions would be like Italian food without tomatoes.

(Somehow the Romans survived those days....)

John Baez (Feb 28 2023 at 23:46):

If you want to organize your thinking about sets, monoids, groups, rings, etc. into a clear picture instead of just a messy mish-mash, you need to realize that these are objects in different categories, and there are adjunctions between these categories.

Peiyuan Zhu (Feb 28 2023 at 23:53):

Ok going back to the Rel category that I'm focusing on. I think I have some interesting questions now. First, if I have a mapping from a set to a power set, or in other words multivalued mapping between two sets, is that a functor? I'm going to try to prove and see (which seems to be true already given your example of topological space). Second it'd be interesting to see if the small and large inverse image is adjoint to the multivalued mapping, which is the question that I posed under the topic Rel category.

Peiyuan Zhu (Feb 28 2023 at 23:55):

Peiyuan Zhu said:

For a multivalued mapping $\Gamma:X\rightarrow Y$, how to undertand its small inverse image $\Gamma^+(U)=\{x\in X:\Gamma(x)\subset U\}$ and large inverse image $\Gamma^-(U)=\{x\in X:\Gamma(x)\cap U\ne\emptyset\}$? Why is this structure important? Is there any categorical meaning behind it?

Just to interface with this topic

John Baez (Feb 28 2023 at 23:55):

First, if I have a mapping from a set to a power set, or in other words multivalued mapping between two sets, is that a functor?

No, though you can turn almost anything into a functor if you try.

John Baez (Feb 28 2023 at 23:57):

It's better to think of this as a morphism in a Kleisli category... if you want to think about it using category theory.

John Baez (Feb 28 2023 at 23:58):

And now we're back to monads, because to understand [[Kleisli categories]] you need [[monads]].

John Baez (Feb 28 2023 at 23:59):

(I'm not doing this to punish you: it's really true! When I think "mapping from a set to a power set" I think "Kleisli category".)

Peiyuan Zhu (Mar 01 2023 at 00:03):

Ok, I'll look into adjoint -> monad -> Kleisli categories

Peiyuan Zhu (Mar 01 2023 at 00:04):

In the book Topoi the word monad only appeared once and it's in a very later chapter on logics. I there an easier entry to this topic?

David Egolf (Mar 01 2023 at 00:09):

Peiyuan Zhu said:

In the book Topoi the word monad only appeared once and it's in a very later chapter on logics. I there an easier entry to this topic?

"Category Theory in Context" (by Riehl) talks about monads in Chapter 5, with Chapter 4 on adjunctions. "Notes on Category Theory" (by Perrone) also has a Chapter 4 on adjunctions and a Chapter 5 on monads. These books still take a lot of work to understand (at least in my experience) though.

Peiyuan Zhu (Mar 01 2023 at 00:10):

Thanks, these are very helpful. It's good to know that it's not a functor... saves my life

Peiyuan Zhu (Mar 01 2023 at 00:18):

John Baez said:

It's better to think of this as a morphism in a Kleisli category... if you want to think about it using category theory.

Is there a reason why in category theory there's this strict hierarchy of natural transformation, functor, morphism and objects?

Peiyuan Zhu (Mar 01 2023 at 00:19):

Peiyuan Zhu said:

Peiyuan Zhu said:

For a multivalued mapping $\Gamma:X\rightarrow Y$, how to undertand its small inverse image $\Gamma^+(U)=\{x\in X:\Gamma(x)\subset U\}$ and large inverse image $\Gamma^-(U)=\{x\in X:\Gamma(x)\cap U\ne\emptyset\}$? Why is this structure important? Is there any categorical meaning behind it?

Just to interface with this topic

This appears to me quite like adjunction, because both are "approximate inverse", but in fact it's not even a functor.

John Baez (Mar 01 2023 at 00:20):

Peiyuan Zhu said:

John Baez said:

It's better to think of this as a morphism in a Kleisli category... if you want to think about it using category theory.

Is there a reason why in category theory there's this strict hierarchy of natural transformation, functor, morphism and objects?

Yes. But I really urge you to read and understand Eugenia Cheng's book The Joy of Abstraction, which explains a lot of this stuff in plain English - including adjunctions.

Peiyuan Zhu (Mar 01 2023 at 00:23):

Ok nice. I love plain English.

John Baez (Mar 01 2023 at 00:23):

It's a lot easier than the books David Egolf just mentioned. Those books are good too, but they should come after Cheng's book.

John Baez (Mar 01 2023 at 00:24):

I guess you've seen here that there's a reading course on Cheng's book happening now at the Topos Institute.

Peiyuan Zhu (Mar 01 2023 at 00:24):

A few years ago I saw her video on gender essentialism and I quite liked it. Didn't know that she's got a book.

Peiyuan Zhu (Mar 01 2023 at 00:24):

John Baez said:

I guess you've seen here that there's a reading course on Cheng's book happening now at the Topos Institute.

No I didn't know. I'll check the website. Also which channel do people post these stuffs?

John Baez (Mar 01 2023 at 00:25):

Read my blog article about it.

John Baez (Mar 01 2023 at 00:25):

Events like this reading course are announced in the channel #general: events.

Peiyuan Zhu (Mar 01 2023 at 00:27):

Ok thanks this is much needed. The book club is only through question submission but not any online discussion like usual book clubs right?

John Baez (Mar 01 2023 at 00:27):

Right.

David Egolf (Mar 01 2023 at 00:28):

By the way, here is the part of Cheng's book which briefly touches on adjunctions and monads:
adjunctions and monads

John Baez (Mar 01 2023 at 00:30):

Oh, I thought she had more on adjunctions but I guess not. She says

Part 2 gets into category theory, including limits and colimits, duality, functors, natural transformations, and the Yoneda lemma.

John Baez (Mar 01 2023 at 00:30):

Anyway, all this stuff is utterly fundamental and fun.

John Baez (Mar 01 2023 at 01:26):

For adjunctions I guess I'd recommend Leinster's free book Basic Category Theory as being easier than Emily Riehl's book. I haven't read Perrone's yet. So, for beginners I tend to say "first read Cheng, then Leinster, then Riehl" - or maybe several at once.

David Egolf (Mar 01 2023 at 01:43):

John Baez said:

For adjunctions I guess I'd recommend Leinster's free book Basic Category Theory as being easier than Emily Riehl's book. I haven't read Perrone's yet. So, for beginners I tend to say "first read Cheng, then Leinster, then Riehl" - or maybe several at once.

Oh, excellent! Maybe this will be easier for me than Riehl or Perrone (which I both quite like, but are slow going).

I notice that Leinster introduces adjoint functors pretty early, which is exciting. Upon looking at Leinster a bit more, I am also happy to see that the examples require less background knowledge than those in Riehl.

Morgan Rogers (he/him) (Mar 02 2023 at 10:04):

Peiyuan Zhu said:

Peiyuan Zhu said:

Peiyuan Zhu said:

For a multivalued mapping $\Gamma:X\rightarrow Y$, how to undertand its small inverse image $\Gamma^+(U)=\{x\in X:\Gamma(x)\subset U\}$ and large inverse image $\Gamma^-(U)=\{x\in X:\Gamma(x)\cap U\ne\emptyset\}$? Why is this structure important? Is there any categorical meaning behind it?

Just to interface with this topic

This appears to me quite like adjunction, because both are "approximate inverse", but in fact it's not even a functor.

You conflated two things here. A map $\Gamma: X \to \mathcal{P}(Y)$ is not inherently a functor. However, if we view powersets as partially ordered sets (which are a special case of categories), then the map $\Gamma_*: \mathcal{P}(X) \to \mathcal{P}(Y)$ sending $A \subseteq X$ to $\bigcup_{x \in A}f(x)$ is an order-preserving function (the corresponding special case of a functor). Observe that $A \subseteq \Gamma^+ \Gamma_*(A)$ and $\Gamma_*\Gamma^+(U) \subseteq U$. Once you've looked into adjunctions some more, see if you can understand why this makes $\Gamma^+$ a right adjoint to $\Gamma_*$. You can also work out what the relationship is between $\Gamma_*$ and $\Gamma^-$.

Max New (Mar 02 2023 at 14:23):

The closest thing I know of to an actual paper on this topic is this: https://link.springer.com/chapter/10.1007/978-94-009-8462-2_6

Peiyuan Zhu (Mar 02 2023 at 19:36):

Woah this is so helpful. Thanks. Blown my mind.

Peiyuan Zhu (Mar 02 2023 at 19:40):

Reminds me Hamlet:

To be, or not to be, that is the question:
Whether 'tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take Arms against a Sea of troubles,
And by opposing end them: to die, to sleep
No more; and by a sleep, to say we end
The heart-ache, and the thousand natural shocks
That Flesh is heir to? 'Tis a consummation
Devoutly to be wished. To die, to sleep,
To sleep, perchance to Dream; aye, there's the rub,
For in that sleep of death, what dreams may come,
When we have shuffled off this mortal coil,
Must give us pause. There's the respect
That makes Calamity of so long life:
For who would bear the Whips and Scorns of time,
The Oppressor's wrong, the proud man's Contumely, [F: poore]
The pangs of dispised Love, the Law’s delay, [F: dispriz’d]
The insolence of Office, and the spurns
That patient merit of th'unworthy takes,
When he himself might his Quietus make
With a bare Bodkin? Who would Fardels bear, [F: these Fardels]
To grunt and sweat under a weary life,
But that the dread of something after death,
The undiscovered country, from whose bourn
No traveller returns, puzzles the will,
And makes us rather bear those ills we have,
Than fly to others that we know not of?
Thus conscience does make cowards of us all,
And thus the native hue of Resolution
Is sicklied o'er, with the pale cast of Thought,
And enterprises of great pitch and moment, [F: pith]
With this regard their Currents turn awry, [F: away]
And lose the name of Action. Soft you now,
The fair Ophelia? Nymph, in thy Orisons
Be all my sins remember'd.

If Hamlet chose one instead of the other then there wouldn't even be a play Hamlet.

Peiyuan Zhu (Mar 25 2023 at 19:57):

Jorge Soto-Andrade said:

fosco said:

I didn't mention Chile without reason ;-)

Let's have a call you Daniele Palombi and I!

Sure! We should keep trying, now in January, which for us at least is somewhat less busy than December!
By the way, regarding the use of category theory in biology, were you aware of this ZA thesis (Durban)
https://www.academia.edu/51583893/Categorical_systems_biology_an_appreciation_of_categorical_arguments_in_cellular_modelling
and this Japanese paper:
Duality between decomposition and gluing: A theoretical biology via adjoint functors
Taichi Haruna a,∗, Yukio-Pegio Gunji a
doi:10.1016/j.biosystems.2007.02.008 ?
The latter remark, as we did too, that Robert Rosen did not get to the point of taking advantage of adjoint functors in his (elementary) categorical approach to relational biology....

This post seems related

Peiyuan Zhu (Mar 25 2023 at 19:59):

It points to paper: https://www.lab.twcu.ac.jp/~tharuna/Haruna_07_biosystems.pdf

Keith Elliott Peterson (Mar 26 2023 at 01:20):

Here's my handwavey explanation:

Every adjunction gives rise not only to a monad but crucially in the "dialectics" story, also gives rise to a comonad, which is dual to a monad. Both monads and comonads give rise to an Eilenberg-Moore category (or dually, to a Kleisli category).

So, a pair of adjoint functors, seen as opposites:

• "unifies" the opposing functors into an adjunction,
• "lifts" up the opposite categories to a higher order Eilenberg-Moore categories,
• and "abolishes" the conflict of the original categories along the adjoint functors.

Though truth be told, Lawvere equated adjoint triples, or adjoint modalities/adjoints of adjoints, not adjoint pairs, with a "dialectic".

Maybe it would be wise to summon @David Corfield to the conversation.

Peiyuan Zhu (Mar 26 2023 at 01:32):

Here's what the reference says: "Natural systems are usually expressed as dynamical systems that contain the temporal dimension explicitly. At first glance category theory seems to be incompatible with the temporal dimension. For example, a composite arrow in a category must exist before the composition. If one attempts to include the temporal dimension in a category, one 2 has to consider the dynamical change of the category (Ehresmann and Vanbremeersch, 1987). Such an approach regards a category as the structural pattern of a concrete system. However, this is not the only way to view a category. One can view a category as an analytical tool for investigating the common properties of certain objects. Here we take this latter point of view. In particular, an adjunction that is independent of the temporal dimension is the primary tool in the following discussion. Analysis in terms of an adjunction can be applicable to any temporally changing object as long as the object belongs to the category on which the adjunction holds."

Keith Elliott Peterson (Mar 26 2023 at 08:02):

Change the enriching category from $\mathrm{Set}$ to something more amenable to temporal processes.

Or take the notion or composition away from equality to an arrow of some kind.

David Corfield (Mar 27 2023 at 08:53):

I doubt I'd be able to say anything here clearer than what's at nLab: adjoint modality. There's further Lawvere exegesis at Aufhebung.

Peiyuan Zhu (May 05 2023 at 17:27):

Reading this paper of adjoint and emergence: https://philarchive.org/archive/ELLAAE

Keith Elliott Peterson (May 05 2023 at 20:55):

Peiyuan Zhu said:

Reading this paper of adjoint and emergence: https://philarchive.org/archive/ELLAAE

David Ellerman is a pretty good read in my opinion. :big_smile: