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:
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