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charset="utf-8" Content-Transfer-Encoding: quoted-printable In the middle of Posina's original message was "Doesn't it immediately foll= ow from this commonplace understanding of category that morphisms of a cate= gory 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 ful= l embedding of C in C^1) in which the morphisms of C are represented as nat= ural transformations. But to call this theorem a consequence of "catness" would appear to be over= kill. Just as Russell didn't need the existence of God as an axiom (to inj= ect the apparently desired note of philosophy into this thread), neither do= es 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 simu= ltaneously 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 imita= tes that of C. Vaughan Pratt On Sun, Oct 29, 2023 at 1:13=E2=80=AFPM Johnathon Taylor > 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 > wrote: Thank you Dr. Taylor for sharing your unvarnished reading :) Thanking you, Yours truly, posina On Sun, Oct 29, 2023 at 11:48=E2=80=AFPM Johnathon Taylor > wrote: > > I don't know about all that. You are going more deep into the philosophic= al 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 t= hat there seems to be less and less of a point. Everything becomes so prete= ntious 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 i= sn't with regards to mathematical language. Neither do I think philosophica= l 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 compar= e 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 ot= herwise and at that point, specifically for a large enough composite integ= er, that statement doesn't help you study the group very much. > > In the same sense, the things you are attempting to study with natural tr= ansformations become divorced of meaning without the context you are workin= g with. > > All the fancy words and terms you used, don't really do anything for unde= rstanding 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 m= athematician and the point of mathematical papers. > > I come away from this and I am not sure you know what you are talking abo= ut 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 compens= ating . You need to focus on thinking about and writing mathematics in a wa= y 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 wi= th flowery language. > > Johnny > > > On Sun, Oct 29, 2023, 6:12 AM Posina Venkata Rayudu > 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-law= vere, >> 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=E2=80=AFPM Ross Street > wrote: >> > >> > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D >> > "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? >> > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D >> > >> > 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 cat= egory >> > theory was to define natural transformation. That required functor, an= d >> > that required category. >> > >> > Ross You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, 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 mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --000000000000365a640608e47d50 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
In the middle of Pos= ina's original message was "Doesn't it immediately follow from this co= mmonplace understanding of category that morphisms of a category are necess= arily natural transformations ..."

My immediate reactio= n at that point was the following.

Theorem.  The f= unctor 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 transforma= tions.

But to call this the= orem a consequence of "catness" would appear to be overkill. = ; Just as Russell didn't need the existence of God as an axiom (to inject t= he 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 m= orphisms incident on that object.

Other than for compo= sition 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=E2=80=AF= PM Johnathon Taylor <jmt240@case.edu<= /a>> wrote:
You are welcome. I am not a Doctor yet just to let you kn= ow.

I was into the musings of trying to understand category t= heory 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 musi= ng 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.&nbs= p;

Thanks,
Johnathon Taylor

On Sun, Oct 29, 2023, 3:34 PM Posina = Venkata Rayudu <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=E2=80=AFPM Johnathon Taylor <jmt240@case.edu<= /a>> wrote:
>
> I don't know about all that. You are going more deep into the philosop= hical world at this point and leaving out of the world of math. At some poi= nt, you get so disillusioned down this train of what language should be use= d 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 rathe= r 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 wha= t isn't with regards to mathematical language. Neither do I think philosoph= ical 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 t= o compare these two things" (things being statements, sets, categories= , etc...). It turns out that natural transformations give a very fundamenta= l 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 "tha= t 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,&= nbsp; specifically for a large enough composite integer, that statement doe= sn'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 wor= king with.
>
> All the fancy words and terms you used, don't really do anything for u= nderstanding mathematics. You are trying too hard to sound smart and you co= me 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 f= ly. It doesn't sound like you are very confident and as though you are comp= ensating . You need to focus on thinking about and writing mathematics in a way that is concise and gives your audi= ence a feeling that you know what you are talking about and don't have to d= epend on entertaining your audience with flowery language.
>
> Johnny
>
>
> On Sun, Oct 29, 2023, 6:12 AM Posina Venkata Rayudu <posinavra= yudu@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 betwe= en
>> mathematical methods, models, theories, and examples, especially f= rom
>> your perspective (as it appears from your response, Lemay ;)
>>
>> I'll also write again after carefully studying Professor Street's<= br> >> presentation, which is about (the elemental?) natural transformati= ons
>> (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<= br> >> effectiveness of mathematics in natural sciences, along the follow= ing
>> 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 acros= s a
>> given category of experiences (planned perceptions) is given in th= e
>> 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 toward= s
>> "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-n= otes-prof-f-w-lawvere,
>> pp. 101-116), through the relation between Cayley (that you allude= d
>> to) and Yoneda (barely a baby-step:
>> https://conceptualmathematics.substack.com/p/monoid ;)
>>
>> On Sun, Oct 29, 2023 at 12:01=E2=80=AFPM Ross Street <ross= .street@mq.edu.au> wrote:
>> >
>> > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D
>> >  "Yoneda showed that maps in any category can be >> > represented as natural transformations" (Lawvere & S= chanuel,
>> > Conceptual Mathematics, p. 378). Isn't this reason enough to = think of
>> > category theory as the theory of naturality?
>> > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D
>> >
>> > That would be like saying group theory is the theory of permu= tations
>> > (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 o= f us have
>> > given similar talks. The goal of the paper considered the fir= st in category
>> > theory was to define natural transformation. That required fu= nctor, and
>> > that required category.
>> >
>> > Ross
 
 
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