I think it is actively harmful to propagate pseudo-mathematical claims like the those, for example, found in the slides of Guest Lecture 1:
>>> It’s touched or greatly influenced all corners of mathematics.
>>> It’s become a gateway to learning mathematics.
And from the audio of the lecture (paraphrasing):
>>> Category theory is the stem cell that differentiates into and lies at the root of all pure mathematics. ["All forms of pure math" is also written on the slides.]
These statements are simply false. The vast majority of pure mathematics research done today does not involve category theory at all, and does not benefit from it. An even greater majority (like 99%+) of mathematics done in industry and in national labs does not involve category theory. Numerical analysis, probability, statistics, partial differential equations, dynamical systems, harmonic analysis, even lots of modern differential geometry – no category theory to be seen!
Want proof? Pick up any introductory graduate textbook, or any major journal in these fields.
Now, if you want to do research in number theory, or algebraic topology, or algebraic geometry – sure, you likely would benefit from categorical thinking. But those fields hardly have a monopoly on pure mathematics. Even in, for example, Hatcher's introductory graduate text on algebraic topology (perhaps the most widely used), category theory is stuck in a small appendix and you can read the entire thing without it, with no real loss [1].
I don't have the energy right now to explain why I find the lecture series misguided more generally, but I want to at least flag these obviously incorrect statements and urge caution.
[And before anyone grabs a PDE book and tells me the use of cohomology groups in certain places means PDE uses category theory, please note that e.g. homological algebraic and category theory are different things.]
[1] OK, I guess you need to know what e.g. a natural transformation is to read some parts of the last chapter, but no one does that in an introductory course anyway. A motivated teacher could easily present the material in such a way that this didn't matter.
Lots of fields are adopting CT. And being taught CT earlier on in the process would provide a useful framework for building upon future knowledge.
Mathematics seems to have 10 different names for the same concept depending on which field you are in. CT provides a common vocabulary.
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.
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 would say this statement is not false. It does seem to lie at the root and is an alternative to set theory which means that mathematics can be perceived with category theory as it's foundation. Whether an actual written CT reformalization has been executed within all the pure mathematical sub fields or whether doing an actual formalization from the perspective of CT will be a benefit at all is another topic all together.
I agree that category theory is not popular. However I feel software engineers should have interest in it as Category theory looks to intersect with an area of software engineering that is currently done ad-hoc style with just gut feelings.
If you squint, Category theory looks to be a formalized theory for the design and complexity of modules. Fully understanding Category theory and how it applies to software could lead to a future where design and architecture is "calculated" rather then "designed."
They might not think of it consciously, but they sure exploit and use properties of sets, rings, groups, and so on all the time in their calculations...
And I'm pretty sure that if I cherry-picked my credits right, the same would have happened in the Phd.
Also, I discovered that there are cranks even in category theory! There was this guy Robert Rosen who tried to apply it to biology, got totally confused, wrote three giant books, and now has a posthumous following. That wasn't something the lecturers were pushing, but it came up among the students.
I watched the first guest lecture, which was very good. I'm not a mathematician, engineer, or a category theorist, but I can apply the formalisms to system architecture instantly.
Abstract, domain-independent formalisms can make ideas harder to understand than domain-specific, concrete examples. With category theory, I'm not seeing examples of the formalism paying off that would justify the endeavor.
I hope that there simply hasn't been enough work done to do a category-theoretic treatment of all of these topics, and that perhaps even more category theory itself needs to be developed so that there are good ways to talk about concepts that are almost-but-not-quite-entirely described or subsumed by category theory.
The alternative is that I'm painfully wrong about what applied category theory aims to be, and that I have a ton of application-specific terms to learn about and won't find a formalization of the sense in which all of these concepts relate.
[1] https://en.wikibooks.org/wiki/Understanding_Darcs/Patch_theo...
[2] https://github.com/trailofbits/graphtage
[3] http://bx-community.wikidot.com
[4] https://github.com/grammarware/bx-parsing
[5] https://en.wikipedia.org/wiki/QVT
[6] http://graphdatamodeling.com/Graph%20Data%20Modeling/GraphQL...
[7] https://neo4j.com/developer/guide-data-modeling/
[8] https://web-cats.gitlab.io/#some-of-the-cats-we-come-across
[9] http://pauillac.inria.fr/~pilkiewi/papers/boomerang-tr.pdf
The problem with CT education IMO is that it cannot be taught at the abstraction level only -- there are too many floating abstractions that people can't anchor to any existing knowledge.
This means CT can only REALLY be understood once you apply it to a domain. The problem is that not many people outside of mathematicians understand the domains that CT is traditionally taught with.
CT for Engineers, CT for Programmers, CT for XYZ, is probably the only viable way CT is going to see wider adoption.
