[Koshkin]

Personally, I wouldn't be that skeptical. By its very nature CT as a foundational theory and is relevant to, and has at least in that sense indeed touched, all corners of mathematics. (Mathematicians had the same skepticism about Set Theory when it first appeared.) Especially the "theoretical" (pure) math. So, sure, "you can read the entire thing without it, with no real loss", but this only says something about the particular textbook and not the subject itself. I assure you, the actual loss, whether you realize it or not, will be very real. Books like Aluffi's Algebra: Chaper 0 have a very good reason behind them. Category Theory is the chapter zero of the modern understanding of mathematics (and not only).

[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.

[btilly]

By its very nature CT as a foundational theory and is relevant to, and has at least in that sense indeed touched, all corners of mathematics.

It comes as no surprise that category theorists make this kind of argument for their own importance. However many corners of mathematics are filled with mathematicians who disagree. Take 100 random people who work in some combination of combinatorics, functional analysis and probability theory. I'd bet that most have never used category theory in a publication. And this doesn't just apply to a few luddites. Consider someone like Terry Tao. He knows some category theory, of course. But you'll have to look long and hard for any paper of his that uses it, or any explanation based on it.

And when you step outside of mathematics to fields that use mathematics heavily, you'll find that applications get harder to find. When you listen to category theorists, you get the impression that category theory is central to programming. Haskell and Scala in particular make good use of category theory. But is that how things work in the real world?

Here is an experiment. Take 100 random working programmers. Ask them if they have ever used category theory to write any programs. You might find 1, probably not 2. Go look at https://www.tiobe.com/tiobe-index/. No programming language in the top 20 even has good support for category theoretical ideas. (The top one that does is Julia at #29.)

Go outside of programming to something like engineering and it becomes even harder to find anyone who thinks that category theory is relevant to their lives.

I came close to a PhD in math, and have multiple papers. My experience is that I needed to learn category theory for some required courses, and otherwise it had no relevance to anything of interest to me. And I do not believe that my experience in that is particularly atypical.

If you disagree, go learn about some fields like numerical analysis, combinatorics, cryptography, and number theory. Sure, for every field you can find evangelicalists who try to apply category theory. Ignore them, find out what the mainstream research uses. Guess what? You WON'T find that people use the language of category theory. You also won't find many practitioners who think that recasting what they are doing in terms of category theory is very useful. You may think that category theory is required to understand those topics, but the people who demonstrably do understand those topics well disagree. I'm going to go with the subject matter experts self-assessment over yours here!

In short, category theory's domination of mathematics is far less sweeping than adherents like you would have us believe.

I get it. From where you stand, you only see and are interested in areas where category theory matters. To you it looks dominant. But that is an illusion. In fact it is extremely similar to another illusion that I discussed in http://www.dtc.umn.edu/~odlyzko/doc/metcalfe.pdf:

Metcalfe’s Law is intuitively appealing, since our personal estimate of the size of a network is based on the uptake of that network among friends and family. Our derived value also varies directly with that metric. We therefore see a linear relationship between the perceived size and value of that network.

In both cases you get a biased view that causes things of personal interest to you to look more universal than they truly are.