Re: Categories vis-a-vis Naturality

[Johnathon Taylor <jmt240@case.edu>]

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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/JjoKCD1vRkCpG3WAIWR2Zg?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 PL
>
> PS. 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!
>
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>



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