[spekcular]

Can you please explain to me how learning category theory is relevant to, or would enhance the work of, a research mathematician who works on, say, harmonic analysis or probability theory?

What is the actual loss to these mathematicians?

Also, your claim that category theory is a foundational theory (in the sense that set theory is) is just mistaken. Homotopy type theory claims to be such a theory, but homotopy type theory should not be identified with category theory more broadly.

[Koshkin]

I wouldn't try, but Helemskii's Lectures And Exercises on Functional Analysis show all this in exquisite detail.

The Convenient Setting of Global Analysis (freely available as a PDF) makes extensive use of the categorical notions and methods.

[spekcular]

This does not answer my question. The material in the first book is all well-known. A standard reference is Reed and Simon's Methods of Modern Mathematical Physics. Categories don't gain you anything there, and in any case it's tangentially relevant to most work in harmonic analysis and probabiility. The second book is not relevant at all.

Just because you can produce books on analysis where someone uses the word "category," does not mean a working mathematician ought to care.

[Koshkin]

(Funny that you mentioned "a working mathematician.")

Anyway, wouldn't it be nice to really understand, on some (higher, admittedly) level, what it is that you are actually doing? Functors and all...

[spekcular]

But this is exactly my objection. Returning to your example, I don't think working through Helemskii's book helps one really understand functional analysis relative to a well-written standard treatment. What interesting problems does this viewpoint permit a probabilist or harmonic analyst to solve that the standard approach does not? What theorems does it enable?

I find it bizarre that you (and others – I don't mean to pick on you) seem to think that a translation to category theoretic language is necessary (and sufficient?) to understand what one is actually doing. Do the many professional mathematicians who prove important theorems in functional analysis today without bothering to learn this language not actually understand what they are doing [0]? But the undergraduate who reads Helemskii does? This seems like an absurd notion of what it means to actually understand a subject.

[0] See, for example, many of the papers noted here: https://en.wikipedia.org/wiki/Invariant_subspace_problem

[Koshkin]

Well, at the very least Category Theory can help keep things organized. Let me quote Tom Leinster:

K-theory and K-homology have become indispensable tools in operator theory; there is even a bivariant functor 𝐾𝐾(−,−) from the category of C-algebras to the category of abelian groups relating the two constructions, and many deep theorems can be subsumed in the assertion that there is a category whose objects are C-algebras and whose morphism spaces are given by 𝐾𝐾(𝐴,𝐵). Cyclic homology and cohomology has also become extremely relevant to the interface between analysis and topology.