Applied Compositional Thinking for Engineers

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.

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.

I do.

Category theory is something that appeals to some mathematicians, and not to others. Those who it doesn't appeal to are likely to wind up in fields like combinatorics, functional analysis, numerical analysis, and so on. If you inflict category theory prematurely on the latter group, people who might have proven quite talented will be driven away from mathematics.

And I feel this quite personally. I left mathematics for other reasons. But still, had I had to deal with category theory first, I'd have never gone into mathematics in the first place.