Re: Categories vis-a-vis Naturality

[Posina Venkata Rayudu <posinavrayudu@gmail.com>]

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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/ro55CxngGkf8NrKGfYVMKM?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/xBItCyoj8PuDV9EjHRZXTV?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


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