This is exactly the kind of false sense of intellectual superiority to avoid. Mathematicians and category theorists don't talk like this. Even the parts of Haskell that have the flavor of category theory aren't really all that much like category theory.
Category theory in itself is pretty useless. Categories are a bit like graphs in that: they are too general.
There are serious mathematical theories using category theory as their language (mostly commutative algebra and whatever it touches) just as there are serious theories using the language of graphs, but in both cases the meat of the math is not in the underlying language.
If you want to research more about this, I can recommend sheaves in geometry and logic by Maclane and Moerdijk. You can also search for "algebraic set theory" and go from there.
And btw, type theory is essentially the same thing. The axioms are IMO uglier and more ad-hoc, but there are (some proven, some conjectured) equivalence results.
What about the following: https://ncatlab.org/nlab/show/foundation+of+mathematics#CFM
Even if the meat of the math is not in the categorical aspects of a theory, the categorical input can be useful - if nothing else, then as a convenient language.
When you’re working with structures (algebra, geometry, topology, etc), sure, you can construct them in the language of categories. But that’s very far from “all of math”.
Category theory is not a great basis for mathematics, but at least it's not inconsistent.
It's certainly very abstract and fundamental, but it's definitely not the root field.
This is disputed and seems to be at best conjecture/intuition without adequate formal definition supporting it.
2. Just because one thing captures something fundamental about computation does not mean it captures everything fundamental about computation, nor does it mean nothing else can capture fundamental aspects of computation (disjoint from what may or may not be captured by category theory). For example, category theory does a terrible job representing the mathematical subfield of complexity theory, despite it being entirely about computation. For that matter, "how abstract" is not always comparable, and the intellectual dick measuring of what is the most abstract is counterproductive. Even basic questions in particle physics have no known model in mathematics that doesn't break under gentle prodding, so is particle physics more abstract than mathematics?
3. Haskell does a lousy job representing the majority of ideas that make category theory useful to mathematicians. For example, universal properties are neglected Haskell, and at best they're implicit. Nobody actually thinks about the math when they're writing Haskell programs, they just think about Haskell types and programs, which makes the "techniques and patterns" of Haskell programming just as arbitrary as the conventions in random Javascript libraries. It's just that there are fewer names for the same thing because fewer people write serious programs in Haskell.
That’s an excellent point. So many people think they’re doing category theory when they’re writing Haskell, but Haskell is essentially nothing like the mathematics it borrows terminology from. They’re really not comparable. Which is absolutely fine, but then people shouldn’t draw parallels to the actual mathematics when the terminology is similar in name only.
Can you expound on this? Just because the type classes defined in Haskell can't represent a wider range of categorical structures they purport to capture, does not mean that more structure doesn't exist throughout a Haskell code base. For instance, many argue that the only type of functors used in Haskell are endofunctors, but thats not really true unless we have a narrow definition of 'functors used in haskell', no?