I don't know about all that. You are going more deep into the philosophical world at this point and leaving out of the world of math. At some point, you get so disillusioned down this train of what language should be used that there seems to be less and less of a point. Everything becomes so pretentious and so above-it-all that it circles back around and becomes nonsense that we can only think about rather than do anything with. Granted, I am not sure this is the appropriate place to post this type of flowery discussion of a philosophical pondering of what is real and what isn't with regards to mathematical language. Neither do I think philosophical musing will make you and less or more proficient at category theory. You can look at all of mathematics at its most basic as "I want to compare these two things" (things being statements, sets, categories, etc...). It turns out that natural transformations give a very fundamental and general view on how to compare two categories which encapsulate alot of important things in mathematics. I think Professor Street said it best, however, when he said "that would be like saying group theory is the theory of permutations". It is true that groups embed into permutations but the image of the imbedding is entirely dependent on the group. You know nothing of the permutations you picked otherwise and at that point, specifically for a large enough composite integer, that statement doesn't help you study the group very much. In the same sense, the things you are attempting to study with natural transformations become divorced of meaning without the context you are working with. All the fancy words and terms you used, don't really do anything for understanding mathematics. You are trying too hard to sound smart and you come away not helping anyone understand what is going on which is the job of a mathematician and the point of mathematical papers. I come away from this and I am not sure you know what you are talking about or if you are trying to sound smart and coming up with stuff on the fly. It doesn't sound like you are very confident and as though you are compensating . You need to focus on thinking about and writing mathematics in a way that is concise and gives your audience a feeling that you know what you are talking about and don't have to depend on entertaining your audience with flowery language. Johnny On Sun, Oct 29, 2023, 6:12 AM Posina Venkata Rayudu <posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>> wrote: Dear Professors: Street, Rosebrugh, Lemay, Taylor et al., Thank you very much for positng my working-question (Lemay :) I'll write to you again after thinking through the relations between mathematical methods, models, theories, and examples, especially from your perspective (as it appears from your response, Lemay ;) I'll also write again after carefully studying Professor Street's presentation, which is about (the elemental?) natural transformations (as in: natural transformation is required to define functor which, in turn, is required to define category). For now, in the spirit of full disclosure, natural transformation, in the sense of structure-respecting maps, appear to account for the effectiveness of mathematics in natural sciences, along the following lines (open to their fate ;) 1. We are given 'change', which we objectify (e.g., physical constrasts (particulars) are sensed by featherless biped brains ;) objects are perceived; geometric objectification of objects as structures is made possible thanks to our minds (mental concepts i.e., properties along with their mutual determinations). 2. Given that a concept (abstract general) that is invariant across a given category of experiences (planned perceptions) is given in the given (change), surely, the given makes it possible to objecfity (the invariant of a category of the given changes). Isn't it yet another reason to reorient science/mathematics towards "the given" and away from its (pathalogical ;) fixation on) "exits" (see Rosebrugh & Lawvere, Sets for Mathematics, p. 240)? I look forward to your corrections (unvarinshed ;) Happy Weekend :) Thanking you, Yours truly, posina P.S. Professor Street, I recently started working my way, inspired by Professor by F. William Lawvere's Perugia Notes (https://conceptualmathematics.substack.com/p/perugia-notes-prof-f-w-lawvere, pp. 101-116), through the relation between Cayley (that you alluded to) and Yoneda (barely a baby-step: https://conceptualmathematics.substack.com/p/monoid ;) On Sun, Oct 29, 2023 at 12:01 PM Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:

================================================ "Yoneda showed that maps in any category can be represented as natural transformations" (Lawvere & Schanuel, Conceptual Mathematics, p. 378). Isn't this reason enough to think of category theory as the theory of naturality? ================================================

That would be like saying group theory is the theory of permutations (because of the Cayley theorem).

Perhaps my little colloquium talk entitled

``The natural transformation in mathematics''

at

http://science.mq.edu.au/~street/MathCollMar2017_h.pdf

would be of some interest in this connexion. I am sure lots of us have given similar talks. The goal of the paper considered the first in category theory was to define natural transformation. That required functor, and that required category.

Ross

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You are welcome. I am not a Doctor yet just to let you know. I was into the musings of trying to understand category theory in this type of abstract sense only a few years ago.

From experience and conversations with people who do have the Doctor title and have years of experience beyond me, this type of musing about abstract theory in this level of abstractness is not particularly interesting and it causes more burnout then trying to work with tetra-categories and a theory involving them.

Thanks, Johnathon Taylor On Sun, Oct 29, 2023, 3:34 PM Posina Venkata Rayudu <posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>> wrote: Thank you Dr. Taylor for sharing your unvarnished reading :) Thanking you, Yours truly, posina On Sun, Oct 29, 2023 at 11:48 PM Johnathon Taylor <jmt240@case.edu<mailto:jmt240@case.edu>> wrote:

I don't know about all that. You are going more deep into the philosophical world at this point and leaving out of the world of math. At some point, you get so disillusioned down this train of what language should be used that there seems to be less and less of a point. Everything becomes so pretentious and so above-it-all that it circles back around and becomes nonsense that we can only think about rather than do anything with.

Granted, I am not sure this is the appropriate place to post this type of flowery discussion of a philosophical pondering of what is real and what isn't with regards to mathematical language. Neither do I think philosophical musing will make you and less or more proficient at category theory.

You can look at all of mathematics at its most basic as "I want to compare these two things" (things being statements, sets, categories, etc...). It turns out that natural transformations give a very fundamental and general view on how to compare two categories which encapsulate alot of important things in mathematics.

I think Professor Street said it best, however, when he said "that would be like saying group theory is the theory of permutations". It is true that groups embed into permutations but the image of the imbedding is entirely dependent on the group. You know nothing of the permutations you picked otherwise and at that point, specifically for a large enough composite integer, that statement doesn't help you study the group very much.

In the same sense, the things you are attempting to study with natural transformations become divorced of meaning without the context you are working with.

All the fancy words and terms you used, don't really do anything for understanding mathematics. You are trying too hard to sound smart and you come away not helping anyone understand what is going on which is the job of a mathematician and the point of mathematical papers.

I come away from this and I am not sure you know what you are talking about or if you are trying to sound smart and coming up with stuff on the fly. It doesn't sound like you are very confident and as though you are compensating . You need to focus on thinking about and writing mathematics in a way that is concise and gives your audience a feeling that you know what you are talking about and don't have to depend on entertaining your audience with flowery language.

Johnny

On Sun, Oct 29, 2023, 6:12 AM Posina Venkata Rayudu <posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>> wrote:

...

Dear Professors: Street, Rosebrugh, Lemay, Taylor et al.,

Thank you very much for positng my working-question (Lemay :)

I'll write to you again after thinking through the relations between mathematical methods, models, theories, and examples, especially from your perspective (as it appears from your response, Lemay ;)

I'll also write again after carefully studying Professor Street's presentation, which is about (the elemental?) natural transformations (as in: natural transformation is required to define functor which, in turn, is required to define category).

For now, in the spirit of full disclosure, natural transformation, in the sense of structure-respecting maps, appear to account for the effectiveness of mathematics in natural sciences, along the following lines (open to their fate ;)

1. We are given 'change', which we objectify (e.g., physical constrasts (particulars) are sensed by featherless biped brains ;) objects are perceived; geometric objectification of objects as structures is made possible thanks to our minds (mental concepts i.e., properties along with their mutual determinations).

2. Given that a concept (abstract general) that is invariant across a given category of experiences (planned perceptions) is given in the given (change), surely, the given makes it possible to objecfity (the invariant of a category of the given changes).

Isn't it yet another reason to reorient science/mathematics towards "the given" and away from its (pathalogical ;) fixation on) "exits" (see Rosebrugh & Lawvere, Sets for Mathematics, p. 240)?

I look forward to your corrections (unvarinshed ;)

Happy Weekend :)

Thanking you, Yours truly, posina P.S. Professor Street, I recently started working my way, inspired by Professor by F. William Lawvere's Perugia Notes (https://conceptualmathematics.substack.com/p/perugia-notes-prof-f-w-lawvere, pp. 101-116), through the relation between Cayley (that you alluded to) and Yoneda (barely a baby-step: https://conceptualmathematics.substack.com/p/monoid ;)

On Sun, Oct 29, 2023 at 12:01 PM Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:

...

================================================ "Yoneda showed that maps in any category can be represented as natural transformations" (Lawvere & Schanuel, Conceptual Mathematics, p. 378). Isn't this reason enough to think of category theory as the theory of naturality? ================================================

That would be like saying group theory is the theory of permutations (because of the Cayley theorem).

Perhaps my little colloquium talk entitled

``The natural transformation in mathematics''

at

http://science.mq.edu.au/~street/MathCollMar2017_h.pdf

would be of some interest in this connexion. I am sure lots of us have given similar talks. The goal of the paper considered the first in category theory was to define natural transformation. That required functor, and that required category.

Ross

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I am enjoying the renewed liveliness of the list. However.... I am reminded of the dictum "all concepts are Kan extensions". It is true, in the same way that "all concepts are terminal objects" .... in a carefully chosen category. In an (infinity, 1)-category people would talk about contractibility of a space of choices. But in my work in category theory I have never explicitly used Kan extensions, whereas I have used limits, colimits, adjoints, Yoneda, naturality (yes) etc. It is this reductionism of all things to a single type of object that can lead to the way set theorists had reduced mathematics to the \in relation. Debates over the "fundamental-ness" of \in vs composition by set theorist logicians and category theorists were not, ultimately, productive, despite philosophical arguments brought to bear by both sides. The renewed love for the list will, I hope, not be dampened, by a discussion of minutiae arguing over this or that philosophical point. I think it a interesting point to ponder, to discuss at a conference, to chat about on the web in more focussed locations. But in a mailing list with presumably hundreds of recipients, it is good to be mindful of not overwhelming all of them with ideas still in the gestational phase. With respect David On Mon, 30 Oct 2023, 6:43 am Posina Venkata Rayudu, <posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>> wrote: CAUTION: External email. Only click on links or open attachments from trusted senders. ________________________________ Thank you Dr. Taylor for sharing your unvarnished reading :) Thanking you, Yours truly, posina On Sun, Oct 29, 2023 at 11:48 PM Johnathon Taylor <jmt240@case.edu<mailto:jmt240@case.edu>> wrote:

I don't know about all that. You are going more deep into the philosophical world at this point and leaving out of the world of math. At some point, you get so disillusioned down this train of what language should be used that there seems to be less and less of a point. Everything becomes so pretentious and so above-it-all that it circles back around and becomes nonsense that we can only think about rather than do anything with.

Granted, I am not sure this is the appropriate place to post this type of flowery discussion of a philosophical pondering of what is real and what isn't with regards to mathematical language. Neither do I think philosophical musing will make you and less or more proficient at category theory.

You can look at all of mathematics at its most basic as "I want to compare these two things" (things being statements, sets, categories, etc...). It turns out that natural transformations give a very fundamental and general view on how to compare two categories which encapsulate alot of important things in mathematics.

I think Professor Street said it best, however, when he said "that would be like saying group theory is the theory of permutations". It is true that groups embed into permutations but the image of the imbedding is entirely dependent on the group. You know nothing of the permutations you picked otherwise and at that point, specifically for a large enough composite integer, that statement doesn't help you study the group very much.

In the same sense, the things you are attempting to study with natural transformations become divorced of meaning without the context you are working with.

All the fancy words and terms you used, don't really do anything for understanding mathematics. You are trying too hard to sound smart and you come away not helping anyone understand what is going on which is the job of a mathematician and the point of mathematical papers.

I come away from this and I am not sure you know what you are talking about or if you are trying to sound smart and coming up with stuff on the fly. It doesn't sound like you are very confident and as though you are compensating . You need to focus on thinking about and writing mathematics in a way that is concise and gives your audience a feeling that you know what you are talking about and don't have to depend on entertaining your audience with flowery language.

Johnny

On Sun, Oct 29, 2023, 6:12 AM Posina Venkata Rayudu <posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>> wrote:

...

Dear Professors: Street, Rosebrugh, Lemay, Taylor et al.,

Thank you very much for positng my working-question (Lemay :)

I'll write to you again after thinking through the relations between mathematical methods, models, theories, and examples, especially from your perspective (as it appears from your response, Lemay ;)

I'll also write again after carefully studying Professor Street's presentation, which is about (the elemental?) natural transformations (as in: natural transformation is required to define functor which, in turn, is required to define category).

For now, in the spirit of full disclosure, natural transformation, in the sense of structure-respecting maps, appear to account for the effectiveness of mathematics in natural sciences, along the following lines (open to their fate ;)

1. We are given 'change', which we objectify (e.g., physical constrasts (particulars) are sensed by featherless biped brains ;) objects are perceived; geometric objectification of objects as structures is made possible thanks to our minds (mental concepts i.e., properties along with their mutual determinations).

2. Given that a concept (abstract general) that is invariant across a given category of experiences (planned perceptions) is given in the given (change), surely, the given makes it possible to objecfity (the invariant of a category of the given changes).

Isn't it yet another reason to reorient science/mathematics towards "the given" and away from its (pathalogical ;) fixation on) "exits" (see Rosebrugh & Lawvere, Sets for Mathematics, p. 240)?

I look forward to your corrections (unvarinshed ;)

Happy Weekend :)

Thanking you, Yours truly, posina P.S. Professor Street, I recently started working my way, inspired by Professor by F. William Lawvere's Perugia Notes (https://conceptualmathematics.substack.com/p/perugia-notes-prof-f-w-lawvere, pp. 101-116), through the relation between Cayley (that you alluded to) and Yoneda (barely a baby-step: https://conceptualmathematics.substack.com/p/monoid ;)

On Sun, Oct 29, 2023 at 12:01 PM Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:

...

================================================ "Yoneda showed that maps in any category can be represented as natural transformations" (Lawvere & Schanuel, Conceptual Mathematics, p. 378). Isn't this reason enough to think of category theory as the theory of naturality? ================================================

That would be like saying group theory is the theory of permutations (because of the Cayley theorem).

Perhaps my little colloquium talk entitled

``The natural transformation in mathematics''

at

http://science.mq.edu.au/~street/MathCollMar2017_h.pdf

would be of some interest in this connexion. I am sure lots of us have given similar talks. The goal of the paper considered the first in category theory was to define natural transformation. That required functor, and that required category.

Ross

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Fruit salad is a "formula". Fruit is a "terms". Tomato is a "sort". or Fruit salad is a "proof". Fruit is a "formula". Tomato is a "term". San Marzano is a "sort". And needless to say, we shouldn't confuse fruit with fruit salad. That's clear. Less clear is confusing fruit and tomato. Very few distinguish between San Marzano and tomato. Best, Patrik On 2023-10-30 00:38, dawson wrote:

I'm with David here.

For some purposes it is genuinely useful to know that all categorical concepts can be reduced to "terminal object", or that the entire theory of deterministic computation can be emulated within group theory. But that doesn't mean that this should always be done! Mathematics is all about knowing many ways to look at something, and choosing the right one(s).

"Knowledge is knowing that a tomato is a fruit. Wisdom is not putting it into a fruit salad."

Best to all, Robert Dawson

----------

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"ideas still in the gestational phase ..." --- This is a very good point. I'm reminded by Shakespeare's "All the world's a stage": Too infant, or too gestational is not good, I tend to agree. But indeed it's up to the moderator. Of course, it's also up to members of FoM to read or not to read, as we like it. Ideas from the "whining schoolboy, with his satchel ... unwilling to school"? Allowed or not? Yes, some, why not, but again, it's up to the moderator. "Then the lover, with a woeful ballad ..." I haven't seen much of that. Maybe good so. Ballad easily turns to sallad, like in FoM. "Then a soldier ... jealous in honor, sudden and quick in quarrel". I've seen those, indeed "seeking the bubble reputation, even in the cannon's mouth". More of that in FoM, I would say, less within the catlist. Maybe FoM is kind of a doglist. "And then the justice, In fair round belly with good capon lined, With eyes severe and beard of formal cut, Full of wise saws and modern instances;" Yes. This is catlist more than FoM. Nobody named. Everybody highly respected. They've all earned it. "The sixth age shifts Into the lean and slippered pantaloon, for men, with spectacles on nose, for women, and pouch on side, for anyone who prefers to look that way". They should speak more, write more. Please do. If not, passing between generations happens over a shorter interval for generations. If that interval restricts to [quick in quarrel, full of wise saws and modern instances], iterative development will be nothing more. Nothing less either. At "Sans teeth, sans eyes, sans taste, sans everything" there is no more, or, there is everything needed for continuous development across generations. --- Best, Patrik On 2023-10-29 23:14, David Roberts wrote: I am enjoying the renewed liveliness of the list. However.... I am reminded of the dictum "all concepts are Kan extensions". It is true, in the same way that "all concepts are terminal objects" .... in a carefully chosen category. In an (infinity, 1)-category people would talk about contractibility of a space of choices. But in my work in category theory I have never explicitly used Kan extensions, whereas I have used limits, colimits, adjoints, Yoneda, naturality (yes) etc. It is this reductionism of all things to a single type of object that can lead to the way set theorists had reduced mathematics to the \in relation. Debates over the "fundamental-ness" of \in vs composition by set theorist logicians and category theorists were not, ultimately, productive, despite philosophical arguments brought to bear by both sides. The renewed love for the list will, I hope, not be dampened, by a discussion of minutiae arguing over this or that philosophical point. I think it a interesting point to ponder, to discuss at a conference, to chat about on the web in more focussed locations. But in a mailing list with presumably hundreds of recipients, it is good to be mindful of not overwhelming all of them with ideas still in the gestational phase. With respect David On Mon, 30 Oct 2023, 6:43 am Posina Venkata Rayudu, <posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>> wrote: CAUTION: External email. Only click on links or open attachments from trusted senders. ________________________________ Thank you Dr. Taylor for sharing your unvarnished reading :) Thanking you, Yours truly, posina On Sun, Oct 29, 2023 at 11:48 PM Johnathon Taylor <jmt240@case.edu<mailto:jmt240@case.edu>> wrote:

I don't know about all that. You are going more deep into the philosophical world at this point and leaving out of the world of math. At some point, you get so disillusioned down this train of what language should be used that there seems to be less and less of a point. Everything becomes so pretentious and so above-it-all that it circles back around and becomes nonsense that we can only think about rather than do anything with.

Granted, I am not sure this is the appropriate place to post this type of flowery discussion of a philosophical pondering of what is real and what isn't with regards to mathematical language. Neither do I think philosophical musing will make you and less or more proficient at category theory.

You can look at all of mathematics at its most basic as "I want to compare these two things" (things being statements, sets, categories, etc...). It turns out that natural transformations give a very fundamental and general view on how to compare two categories which encapsulate alot of important things in mathematics.

I think Professor Street said it best, however, when he said "that would be like saying group theory is the theory of permutations". It is true that groups embed into permutations but the image of the imbedding is entirely dependent on the group. You know nothing of the permutations you picked otherwise and at that point, specifically for a large enough composite integer, that statement doesn't help you study the group very much.

In the same sense, the things you are attempting to study with natural transformations become divorced of meaning without the context you are working with.

All the fancy words and terms you used, don't really do anything for understanding mathematics. You are trying too hard to sound smart and you come away not helping anyone understand what is going on which is the job of a mathematician and the point of mathematical papers.

I come away from this and I am not sure you know what you are talking about or if you are trying to sound smart and coming up with stuff on the fly. It doesn't sound like you are very confident and as though you are compensating . You need to focus on thinking about and writing mathematics in a way that is concise and gives your audience a feeling that you know what you are talking about and don't have to depend on entertaining your audience with flowery language.

Johnny

On Sun, Oct 29, 2023, 6:12 AM Posina Venkata Rayudu <posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>> wrote:

...

Dear Professors: Street, Rosebrugh, Lemay, Taylor et al.,

Thank you very much for positng my working-question (Lemay :)

I'll write to you again after thinking through the relations between mathematical methods, models, theories, and examples, especially from your perspective (as it appears from your response, Lemay ;)

I'll also write again after carefully studying Professor Street's presentation, which is about (the elemental?) natural transformations (as in: natural transformation is required to define functor which, in turn, is required to define category).

For now, in the spirit of full disclosure, natural transformation, in the sense of structure-respecting maps, appear to account for the effectiveness of mathematics in natural sciences, along the following lines (open to their fate ;)

1. We are given 'change', which we objectify (e.g., physical constrasts (particulars) are sensed by featherless biped brains ;) objects are perceived; geometric objectification of objects as structures is made possible thanks to our minds (mental concepts i.e., properties along with their mutual determinations).

2. Given that a concept (abstract general) that is invariant across a given category of experiences (planned perceptions) is given in the given (change), surely, the given makes it possible to objecfity (the invariant of a category of the given changes).

Isn't it yet another reason to reorient science/mathematics towards "the given" and away from its (pathalogical ;) fixation on) "exits" (see Rosebrugh & Lawvere, Sets for Mathematics, p. 240)?

I look forward to your corrections (unvarinshed ;)

Happy Weekend :)

Thanking you, Yours truly, posina P.S. Professor Street, I recently started working my way, inspired by Professor by F. William Lawvere's Perugia Notes (https://conceptualmathematics.substack.com/p/perugia-notes-prof-f-w-lawvere, pp. 101-116), through the relation between Cayley (that you alluded to) and Yoneda (barely a baby-step: https://conceptualmathematics.substack.com/p/monoid ;)

On Sun, Oct 29, 2023 at 12:01 PM Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:

...

================================================ "Yoneda showed that maps in any category can be represented as natural transformations" (Lawvere & Schanuel, Conceptual Mathematics, p. 378). Isn't this reason enough to think of category theory as the theory of naturality? ================================================

That would be like saying group theory is the theory of permutations (because of the Cayley theorem).

Perhaps my little colloquium talk entitled

``The natural transformation in mathematics''

at

http://science.mq.edu.au/~street/MathCollMar2017_h.pdf

would be of some interest in this connexion. I am sure lots of us have given similar talks. The goal of the paper considered the first in category theory was to define natural transformation. That required functor, and that required category.

Ross

You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>

Dear Professor Coecke, Thank you very much for sharing with us all your fascinating advances in applying category theory! With all due respect, it's your commitment to pragmatism (as you are immediate concern) that we parts ways; but it's in this spirt of science: https://www.nature.com/articles/s42256-023-00703-8 Thanking you, Yours truly, posina P.S. If only there were a view-from-nowhere ;) On Sun, Oct 29, 2023 at 6:30 PM Bob Coecke <Bob.Coecke@quantinuum.com<mailto:Bob.Coecke@quantinuum.com>> wrote: I think that we have been quite successful in doing stuff with category theory, having made it widely used in quantum industry, by companies like IBM, Google, ourselves, and may others. Photonic quantum computing now uses string diagrams as their main language. Soon we'll report on an educational experiment we did over the summer, showing that category theory can actually make inaccessible things accessible, and likely will affect how physics is thought in secondary school. We even got Jim Lambek into Forbes! 🙂 (the last place they expected to see him one of his sons told me) https://www.forbes.com/sites/moorinsights/2021/10/13/cambridge-quantum-makes... [https://imageio.forbes.com/specials-images/imageserve/61672daae1c3ad03b91e3a1c/0x0.jpg?format=jpg&height=900&width=1600&fit=bounds] Cambridge Quantum Makes Quantum Natural Language Processing A Reality Senior Analyst, AI & Quantum Computing, Paul Smith-Goodson, dives in as Cambridge Quantum ("CQ") today announced a quantum computing first. It released an open-source toolkit and library for a Quantum Natural Language Processing (QNLP) toolkit called lambeq. www.forbes.com https://www.forbes.com/sites/moorinsights/2022/04/12/quantinuum-enhances-the... [https://imageio.forbes.com/specials-images/imageserve/6255cb7fe0f6c59460404a9c/0x0.jpg?format=jpg&height=900&width=1600&fit=bounds] Quantinuum Enhances The World’s First Quantum Natural Language Processing Toolkit Making It Even More Powerful Senior Analyst, AI & Quantum Computing, Paul Smith-Goodson, dives in as Quantinuum is an integrated software-hardware quantum computing company that uses trapped-ion for its compute technology. It recently released a significant update to its Lambeq open-source Python library and toolkit. www.forbes.com Honestly...naturality, adjunctions, Yoneda...have never played any direct role in anything we did. Our starting point is that monoidal categories are the direct description of processes, something that first started in computer science, and we then carried over to physics, and then cognition and AI. I'd say that it enables an Heraclites+Schrodinger picture of reality. But there are of course a ton of other uses of CT, which are completely unrelated, if not orthogonal, and rely on naturality, adjunctions, Yoneda... Dogmatic slogans about what category theory is supposed to be have done nothing but damage to the field. It seems to me that the culture has become more healthy and tolerant now, with the growing use of CT in many areas outside of mathematics, where not ideology, but use matters. Cheers, Bob. ________________________________ From: Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> Sent: 29 October 2023 06:31 To: JS Lemay <js.lemay@mq.edu.au<mailto:js.lemay@mq.edu.au>> Cc: Categories mailing list <categories@mq.edu.au<mailto:categories@mq.edu.au>>; posinavrayudu <posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>> Subject: [EXT] Re: Categories vis-a-vis Naturality CAUTION: This email originated from outside of the organization. Do not click links or open attachments unless you recognize the sender and know the content is safe. ================================================ "Yoneda showed that maps in any category can be represented as natural transformations" (Lawvere & Schanuel, Conceptual Mathematics, p. 378). Isn't this reason enough to think of category theory as the theory of naturality? ================================================ That would be like saying group theory is the theory of permutations (because of the Cayley theorem). Perhaps my little colloquium talk entitled ``The natural transformation in mathematics'' at http://science.mq.edu.au/~street/MathCollMar2017_h.pdf would be of some interest in this connexion. I am sure lots of us have given similar talks. The goal of the paper considered the first in category theory was to define natural transformation. That required functor, and that required category. 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