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Email Gateway (Oct 28 2023 at 20:47):From: JS Lemay <js.lemay@mq.edu.au>
From: JS Lemay <js.lemay@mq.edu.au>
Sent: Sunday, October 29, 2023 7:46 AM
To: Categories mailing list <categories@mq.edu.au>
Cc: posinavrayudu <posinavrayudu@gmail.com>
Subject: Categories vis-a-vis Naturality
[[The following email is sent on behalf of posinavrayudu@gmail.com]]
Dear All,
Consider a category of objects (say, cats). Going by a naive
understanding of the notion of category, every object of a category
partakes in an essence/abstract general/theory that is characteristic
of the category (cf. catness; whatever that might be). Doesn't it
immediately follow from this commonplace understanding of category
that morphisms of a category are necessarily natural transformations
preserving the abstract essence characterizing the category (cf.
playful cat ---> pensive cat). Of course, this is an informal
paraphrasing of "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?
Happy Monday :)
Thanking you,
Yours truly,
posina
P.S. Given that all morphisms, beginning with functions between sets
(https://conceptualmathematics.wordpress.com/2022/08/30/functions-are-
natural-
transformations/),
can be construed as natural transformations, one added value of
baptizing category theory as the theory of naturality is that it
brings into figural salience for all see that the constrasting notion
is unnatural (as in miracles) and not social or culture (since they
too don't change willy-nilly). To be clear, highlighting natural is
not intended to belittle the categorical nature of our everyday
experience (Kandel et al., Principles of Neural Science, pp. 621-637).
On Mon, Oct 23, 2023 at 11:10 AM JS Lemay <js.lemay@mq.edu.au> wrote:
Hello everyone!
Welcome to the new category theory mailing list.
Unfortunately, the old categories mailing list categories@mta.ca has gone
permanently offline.Bob Rosebrugh, the old mailing list's moderator, reached out to me and asked
me to set up a new mailing list. So thank you for your patience while we
were getting this new mailing list up and running. (Big thanks to Richard,
Steve, and Macquarie's IT team for helping me with this!)You are receiving this email because you were subscribed to the old
categories mailing categories@mta.ca, and have been added as a member to
this new one.All members from the old mailing list have been added, so this new mailing
list is ready to go! Please feel free to start using this mailing list to
advertise job postings, conferences, call for papers, questions, etc.Also, after 33 years of moderating the mailing list, Bob will be stepping
down from mailing list responsibilities. So a big thank you to Bob for all
these years running the mailing list!A general request email for the mailing list will soon be set up. In the
meantime, if you have any questions regarding the mailing list, or would
like to be removed, or know a colleague who would like to be added, feel
free to email me at js.lemay@mq.edu.auHave a great day!
JS PLPS. We will try to send out emails from the old mailing list that we weren't
sent out before it went offline. Unfortunately, some might be outdated.
Apologies!
Email Gateway (Oct 28 2023 at 21:38):From: JS Lemay <js.lemay@mq.edu.au>
From: Johnathon Taylor <jmt240@case.edu>
Sent: Sunday, October 29, 2023 8:37:38 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: JS Lemay <js.lemay@mq.edu.au>; Categories mailing list <categories@mq.edu.au>; posinavrayudu <posinavrayudu@gmail.com>
Subject: Re: Categories vis-a-vis Naturality
This is not the way that one should consider basic introductory category theory. It should be thought about as the fundamental tool. All the things we naturally care to do in basic category theory has natural transformations in the definition itself.
An equivalence of categories, of which isomorphism is an example of, is defined using natural transformations.
Adjunctions, monads, Monoidal categories, Kan extensions, etc... all are defined using natural transformations.
Naturality is one of the tools used but there are other things we care about and need to be considered.
On Sat, Oct 28, 2023, 5:34 PM Johnathon Taylor <jmt240@case.edu<mailto:jmt240@case.edu>> wrote:
Sorry for that. I had it tagged I thought originally and then had to go fix something.
Thanks,
Johnathon
On Sat, Oct 28, 2023, 5:29 PM JS Lemay <js.lemay@mq.edu.au<mailto:js.lemay@mq.edu.au>> wrote:
Hi Johnathon,
Your email was only sent to myself.
If you’d like to send this reply to the new mailing list: please use categories@mq.edu.au<mailto:categories@mq.edu.au>
and include posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com> as well.
Thanks!
Have a great weekend
JS PL
On Oct 29, 2023, at 8:25 AM, Johnathon Taylor <jmt240@case.edu<mailto:jmt240@case.edu>> wrote:
This is not the way that one should consider basic introductory category theory. It should be thought about as the fundamental tool. All the things we naturally care to do in basic category theory has natural transformations in the definition itself.
An equivalence of categories, of which isomorphism is an example of, is defined using natural transformations.
Adjunctions, monads, Monoidal categories, Kan extensions, etc... all are defined using natural transformations.
Naturality is one of the tools used but there are other things we care about and need to be considered.
On Sat, Oct 28, 2023, 4:47 PM JS Lemay <js.lemay@mq.edu.au<mailto:js.lemay@mq.edu.au>> wrote:
[[The following email is sent on behalf of posinavrayudu@gmail.com<mailto:posinavrayudu@gmail.com>]]
Dear All,
Consider a category of objects (say, cats). Going by a naive
understanding of the notion of category, every object of a category
partakes in an essence/abstract general/theory that is characteristic
of the category (cf. catness; whatever that might be). Doesn't it
immediately follow from this commonplace understanding of category
that morphisms of a category are necessarily natural transformations
preserving the abstract essence characterizing the category (cf.
playful cat ---> pensive cat). Of course, this is an informal
paraphrasing of "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?
Happy Monday :)
Thanking you,
Yours truly,
posina
P.S. Given that all morphisms, beginning with functions between sets
(https://conceptualmathematics.wordpress.com/2022/08/30/functions-are-natural-transformations/<https://protect-au.mimecast.com/s/hSdTCBNqgBCq28k5CzFUBd?domain=conceptualmathematics.wordpress.com>),
can be construed as natural transformations, one added value of
baptizing category theory as the theory of naturality is that it
brings into figural salience for all see that the constrasting notion
is unnatural (as in miracles) and not social or culture (since they
too don't change willy-nilly). To be clear, highlighting natural is
not intended to belittle the categorical nature of our everyday
experience (Kandel et al., Principles of Neural Science, pp. 621-637).
On Mon, Oct 23, 2023 at 11:10 AM JS Lemay <js.lemay@mq.edu.au<mailto:js.lemay@mq.edu.au>> wrote:
Hello everyone!
Welcome to the new category theory mailing list.
Unfortunately, the old categories mailing list categories@mta.ca<mailto:categories@mta.ca> has gone permanently offline.
Bob Rosebrugh, the old mailing list's moderator, reached out to me and asked me to set up a new mailing list. So thank you for your patience while we were getting this new mailing list up and running. (Big thanks to Richard, Steve, and Macquarie's IT team for helping me with this!)
You are receiving this email because you were subscribed to the old categories mailing categories@mta.ca<mailto:categories@mta.ca>, and have been added as a member to this new one.
All members from the old mailing list have been added, so this new mailing list is ready to go! Please feel free to start using this mailing list to advertise job postings, conferences, call for papers, questions, etc.
Also, after 33 years of moderating the mailing list, Bob will be stepping down from mailing list responsibilities. So a big thank you to Bob for all these years running the mailing list!
A general request email for the mailing list will soon be set up. In the meantime, if you have any questions regarding the mailing list, or would like to be removed, or know a colleague who would like to be added, feel free to email me at js.lemay@mq.edu.au<mailto:js.lemay@mq.edu.au>
Have a great day!
JS PLPS. We will try to send out emails from the old mailing list that we weren't sent out before it went offline. Unfortunately, some might be outdated. Apologies!
Email Gateway (Oct 29 2023 at 06:31):From: JS Lemay <js.lemay@mq.edu.au>
From: Ross Street <ross.street@mq.edu.au>
Sent: Sunday, October 29, 2023 5:31:43 PM (UTC+10:00) Canberra, Melbourne, Sydney
To: JS Lemay <js.lemay@mq.edu.au>
Cc: Categories mailing list <categories@mq.edu.au>; posinavrayudu <posinavrayudu@gmail.com>
Subject: Re: Categories vis-a-vis Naturality
================================================
"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
Email Gateway (Oct 29 2023 at 10:12):From: JS Lemay <js.lemay@mq.edu.au>
From: Posina Venkata Rayudu <posinavrayudu@gmail.com>
Sent: Sunday, October 29, 2023 9:12:10 PM (UTC+10:00) Canberra, Melbourne, Sydney
To: Ross Street <ross.street@mq.edu.au>; jmt240 <jmt240@case.edu>
Cc: JS Lemay <js.lemay@mq.edu.au>; Categories mailing list <categories@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
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 ;)
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).
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<https://protect-au.mimecast.com/s/ulOjCyoj8PuDV9E9IQBi2j?domain=conceptualmathematics.substack.com>,
pp. 101-116), through the relation between Cayley (that you alluded
to) and Yoneda (barely a baby-step:
https://conceptualmathematics.substack.com/p/monoid<https://protect-au.mimecast.com/s/oOtsCzvkmpfDVJvJIKwhKJ?domain=conceptualmathematics.substack.com> ;)
On Sun, Oct 29, 2023 at 12:01 PM Ross Street <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
Email Gateway (Oct 29 2023 at 18:19):From: JS Lemay <js.lemay@mq.edu.au>
From: Johnathon Taylor <jmt240@case.edu>
Sent: Monday, October 30, 2023 5:18:29 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: posinavrayudu <posinavrayudu@gmail.com>; Ross Street <ross.street@mq.edu.au>
Cc: Categories mailing list <categories@mq.edu.au>; JS Lemay <js.lemay@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
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 ;)
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).
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<https://protect-au.mimecast.com/s/Y_5dCNLJxkiLy2ZLcmU93A?domain=conceptualmathematics.substack.com>,
pp. 101-116), through the relation between Cayley (that you alluded
to) and Yoneda (barely a baby-step:
https://conceptualmathematics.substack.com/p/monoid<https://protect-au.mimecast.com/s/4UapCOMK7YcYQ6NYiv4Wub?domain=conceptualmathematics.substack.com> ;)
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
Email Gateway (Oct 29 2023 at 19:34):From: JS Lemay <js.lemay@mq.edu.au>
From: Posina Venkata Rayudu <posinavrayudu@gmail.com>
Sent: Monday, October 30, 2023 6:33:45 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: jmt240 <jmt240@case.edu>
Cc: Ross Street <ross.street@mq.edu.au>; Categories mailing list <categories@mq.edu.au>; JS Lemay <js.lemay@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
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> 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> 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 ;)
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).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<https://protect-au.mimecast.com/s/wdX9CQnM1WfVZ52WUMLGaL?domain=conceptualmathematics.substack.com>,
pp. 101-116), through the relation between Cayley (that you alluded
to) and Yoneda (barely a baby-step:
https://conceptualmathematics.substack.com/p/monoid<https://protect-au.mimecast.com/s/x0kLCRONg6sYZAMLSOjrke?domain=conceptualmathematics.substack.com> ;)On Sun, Oct 29, 2023 at 12:01 PM Ross Street <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
Email Gateway (Oct 29 2023 at 19:51):From: JS Lemay <js.lemay@mq.edu.au>
From: Johnathon Taylor <jmt240@case.edu>
Sent: Monday, October 30, 2023 6:50:49 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: posinavrayudu <posinavrayudu@gmail.com>
Cc: Ross Street <ross.street@mq.edu.au>; Categories mailing list <categories@mq.edu.au>; JS Lemay <js.lemay@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
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 ;)
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).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<https://protect-au.mimecast.com/s/1tkHClx1OYUG4j4MiGysl8?domain=conceptualmathematics.substack.com>,
pp. 101-116), through the relation between Cayley (that you alluded
to) and Yoneda (barely a baby-step:
https://conceptualmathematics.substack.com/p/monoid<https://protect-au.mimecast.com/s/2oOCCmO5wZsX494MHOU2o-?domain=conceptualmathematics.substack.com> ;)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
Email Gateway (Oct 29 2023 at 21:16):From: JS Lemay <js.lemay@mq.edu.au>
From: David Roberts <droberts.65537@gmail.com>
Sent: Monday, October 30, 2023 8:14:55 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: Categories mailing list <categories@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
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 ;)
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).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<https://protect-au.mimecast.com/s/iUU_C0YKgRsvPzKRhwBnFW?domain=conceptualmathematics.substack.com>,
pp. 101-116), through the relation between Cayley (that you alluded
to) and Yoneda (barely a baby-step:
https://conceptualmathematics.substack.com/p/monoid<https://protect-au.mimecast.com/s/5h5yCgZ05Jf6QJrLsoScIM?domain=conceptualmathematics.substack.com> ;)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.
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Email Gateway (Oct 29 2023 at 22:58):From: JS Lemay <js.lemay@mq.edu.au>
From: dawson <dawson@cs.smu.ca>
Sent: Monday, October 30, 2023 9:38:03 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: Categories mailing list
Subject: Re: Categories vis-a-vis Naturality
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
Email Gateway (Oct 30 2023 at 00:55):From: JS Lemay <js.lemay@mq.edu.au>
From: Vaughan Pratt <pratt@cs.stanford.edu>
Sent: Monday, October 30, 2023 11:54:29 AM (UTC+10:00) Canberra, Melbourne, Sydney
To: jmt240 <jmt240@case.edu>
Cc: posinavrayudu <posinavrayudu@gmail.com>; Ross Street <ross.street@mq.edu.au>; Categories mailing list <categories@mq.edu.au>; JS Lemay <js.lemay@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
In the middle of Posina's original message was "Doesn't it immediately follow from this commonplace understanding of category that morphisms of a category are necessarily natural transformations ..."
My immediate reaction at that point was the following.
Theorem. The functor category C^1 is a representation of C (that is, a full embedding of C in C^1) in which the morphisms of C are represented as natural transformations.
But to call this theorem a consequence of "catness" would appear to be overkill. Just as Russell didn't need the existence of God as an axiom (to inject the apparently desired note of philosophy into this thread), neither does this theorem need anything more than preservation of units (in this case the unit of 1's only object) and that the unit at each object of C is simultaneously a left and right unit for composition with morphisms incident on that object.
Other than for composition with units, composition at an object of C needn't be associative, or even to exist, just so long as C^1's composition imitates that of C.
Vaughan Pratt
On Sun, Oct 29, 2023 at 1:13 PM Johnathon Taylor <jmt240@case.edu<mailto:jmt240@case.edu>> wrote:
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 ;)
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).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<https://protect-au.mimecast.com/s/ygXyCYW86EsYBx4OS0iP05?domain=conceptualmathematics.substack.com>,
pp. 101-116), through the relation between Cayley (that you alluded
to) and Yoneda (barely a baby-step:
https://conceptualmathematics.substack.com/p/monoid<https://protect-au.mimecast.com/s/zMP3CZY146svBNp2ijLG1Y?domain=conceptualmathematics.substack.com> ;)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<https://protect-au.mimecast.com/s/lSbGC1WLjwsgWY95SpW7Mu?domain=outlook.office365.com> | Leave group<https://protect-au.mimecast.com/s/Lr45C2xMRkUryw9OC2RIy3?domain=outlook.office365.com> | Learn more about Microsoft 365 Groups<https://protect-au.mimecast.com/s/N6DKC3QNl1S3NQE0fQsTSM?domain=aka.ms>
Email Gateway (Oct 30 2023 at 08:01):From: JS Lemay <js.lemay@mq.edu.au>
From: Patrik Eklund <peklund@cs.umu.se>
Sent: Monday, October 30, 2023 6:17:57 PM (UTC+10:00) Canberra, Melbourne, Sydney
To: dawson <dawson@cs.smu.ca>
Cc: Categories mailing list <categories@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
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
You're receiving this message because you're a member of the
Categories mailing list group from Macquarie University.
Email Gateway (Oct 30 2023 at 10:48):From: JS Lemay <js.lemay@mq.edu.au>
From: Patrik Eklund <peklund@cs.umu.se>
Sent: Monday, October 30, 2023 6:08:55 PM (UTC+10:00) Canberra, Melbourne, Sydney
To: droberts.65537@gmail.com <droberts.65537@gmail.com>
Cc: Categories mailing list <categories@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
"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 ;)
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).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<https://protect-au.mimecast.com/s/E3oDCNLJxkiLm5RXumDDaV?domain=conceptualmathematics.substack.com>,
pp. 101-116), through the relation between Cayley (that you alluded
to) and Yoneda (barely a baby-step:
https://conceptualmathematics.substack.com/p/monoid<https://protect-au.mimecast.com/s/vJ48COMK7YcYmrG4CvGTKh?domain=conceptualmathematics.substack.com> ;)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 bec
[message truncated]
Email Gateway (Oct 30 2023 at 10:48):From: JS Lemay <js.lemay@mq.edu.au>
From: ptj@maths.cam.ac.uk <ptj@maths.cam.ac.uk>
Sent: Monday, October 30, 2023 9:40:36 PM (UTC+10:00) Canberra, Melbourne, Sydney
To: dawson <dawson@cs.smu.ca>
Cc: Categories mailing list <categories@mq.edu.au>
Subject: Re: Categories vis-a-vis Naturality
I agree. In my first-year graduate course on category theory, after I've
shown that the three concepts terminal object', right adjoint' and
limit' are all interdefinable, I jokingly add You might say that
category-theorists have only ever had one good idea, and all we do
is to keep dressing it up in new clothes'. (The one good idea' is of
course the notion of universal element, which comes from Yoneda.)
But the dressing up in new clothes' does matter: the introduction of
(appropriate!) new concepts is an important aid to understanding. So
I think it is selling category theory short' to describe it as just'
the study of naturality.
Peter Johnstone
On Oct 29 2023, 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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