Re: Categories vis-a-vis Naturality

[David Roberts <droberts.65537@gmail.com>]

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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<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


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