I agree, the article does not actually present anything concrete. For me, the usefulness of category theory is in providing a simplifying and high level set of abstractions that can serve to illuminate the structural essence of a problem domain. At first, I thought it was just abstract nonsense (and this is coming from someone who has a PhD in mathematics), but after playing around with it a bit, I've started to appreciate the power of the concepts.

The most interesting "application" of category theory that I've seen (as it relates to software engineering) is the connection between databases and category theory, e.g at https://johncarlosbaez.wordpress.com/2018/06/06/applied-cate.... In short, relational schemas are categories, and common data migration operations can very naturally can be described in the language of category theory (functors, pullbacks etc).

> The most interesting "application" of category theory that I've seen (as it relates to software engineering) is the connection between databases and category theory

So is this application productive? It's nice to be able to describe something that already exists in terms of category theory, and even nicer if those descriptions are concise and satisfying. But does that description enable anything new? Did we get any new migration operations, normal forms, etc?

Yes, for example the dual of join (called 'sigma'), which only exists for databases in categorical normal form. There are many academic papers and an open-source project ('categorical query language') here: http://www.categoricaldata.net.

Sounds interesting! I had a look, but couldn't find out what sigma does. This is not quite the same level as, say, the PostgreSQL documentation:

http://www.categoricaldata.net/help/instancesigma.html

Is there somewhere i can go for a worked example of using sigma?

Sure, here is an example that uses sigma to compute an equivalence relation:

http://cql.conexus.ai/quotient.php

FWIW, I think the best "applications" (if they can be called that) come from domains like algebraic topology and algebraic geometry, where some concepts come up so frequently that it's useful to formalize them in category theory. The homology proof of Brouwer's fixed point theorem [1] is one good example. Things like Eilenberg-Maclane spaces are another (really is most natural to think of them the thing that represents some functor).

I wouldn't say that you can't do this stuff without category theory, but it does make it easier / clearer. (Similarly, you can do a lot of geometry without coordinates, but coordinates definitely make some stuff easier / clearer.)

[1] https://www.wikiwand.com/en/Brouwer_fixed-point_theorem#/A_p... [2] https://www.wikiwand.com/en/Eilenberg%E2%80%93MacLane_space

Here's an example. It's not mentioned in the OP, but in a link in the OP. Microsoft's LINQ was designed using principles from category theory. They didn't use category terminology in the final product, and rightfully so.

> Gives three examples of bits of mathematics with really strong concrete uses.

The point, I think, is not that those fields of mathematics have obvious concrete applications to computers, but that they support each other in how they are applied to computer problems. Like, one reason numerical analysis is useful is that it helps you better solve differential equations, a common way to model a linear system. So linear algebra, of course, has these great applications in computers, but they're still pretty indirect because it's not like we program computers by feeding them linear maps (or whatever).

Yup, no applications provided. Only here would this title be clickbait, but here we are. I would suggest "Category theory may inspire results in other fields", but specific example of that is also not present.