I agree. In my first-year graduate course on category theory, after I've shown that the three concepts `terminal object', `right adjoint' and `limit' are all interdefinable, I jokingly add `You might say that category-theorists have only ever had one good idea, and all we do is to keep dressing it up in new clothes'. (The `one good idea' is of course the notion of universal element, which comes from Yoneda.) But the `dressing up in new clothes' does matter: the introduction of (appropriate!) new concepts is an important aid to understanding. So I think it is `selling category theory short' to describe it as `just' the study of naturality. Peter Johnstone On Oct 29 2023, dawson wrote:
I'm with David here.
For some purposes it is genuinely useful to know that all categorical concepts can be reduced to "terminal object", or that the entire theory of deterministic computation can be emulated within group theory. But that doesn't mean that this should always be done! Mathematics is all about knowing many ways to look at something, and choosing the right one(s).
"Knowledge is knowing that a tomato is a fruit. Wisdom is not putting it into a fruit salad."
Best to all, Robert Dawson
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