Category theory is just a special case of edge-labelled digraph theory

Andrej Bauer raises a question whether Hask is a real category. I think it’s a legitimate question to ask, especially by a mathematician or programming languages researcher. But I want to look closer at how a (probably negative) answer to this question would affect Haskell and its community.

nLab is to collect the stable bits, organized, hyperlinked, usefully. ​ Such a far-flung structure is highly vulnerable and, once broken, impossible to restore.

one that emphasises the study of a whole through not only its parts, but the precise way it is built up from its parts. ​ We will welcome papers on implementation.

Category theory and intelligence seem to be deeply linked - both are guided by the goal of organizing and layering abstractions. Automation of exactly that - organizing and layering abstractions - seems to be the quest of machine learning.

I'm excited to share the positive news and to be working on such an interesting project! I believe we have just begun to scratch the surface of applying category theory to understanding the universe around us and inside us. https://t.co/KgpHpkfEgA— Bruno Gavranović (@bgavran3) April 23, 2019

Bartosz Milewski's 'Category Theory for Programmers' unofficial PDF and LaTeX source - GitHub - hmemcpy/milewski-ctfp-pdf: Bartosz Milewski's 'Category Theor...

“Category Theory is the greatest refactor of mathematics in history.”

“Look up pullbacks in differential geometry and you will find the image on the left. Look up pullbacks in category theory and you will find the image on the right, which generalizes everything in one diagram. That is the type of generality that libraries should strive for.”

You see, it's very different than other branches of math. Rather than being another sibling lined up in the family photograph, it's more like a common gene that unites them in the first place. With this vantage point, it becomes evident that different areas of math share common patterns/trends/structures. This becomes extraordinarily useful when you want to solve a problem in one realm (say, topology) but don't have the right tools at your disposal. By transporting the problem to a different realm (say, algebra), you can see the problem in a different light and perhaps discover new tools, and the solution may become much easier. a "template" for all of mathematics: depending on what you feed into the template, you'll recover one of the mathematical realms. Naturally, then, you're prompted to also ask about relationships between categories. These are called functors. But why stop there? What about the relationships between those relationships? These are called natural transformations. (And yes, you can ask a hierarchy of questions: "What about the relationships between the relationships between the relationships between the...?" This leads to infinity categories. [And a possible brain freeze.]

reasoning about the system should be done recursively on its structure. ​ good software design is ultimately an art. ​ another example of reasoning via an interface. ​ I suspect that interfaces are in fact synonymous with compositionality. That is, compositionality is not just the ability to compose objects, but the ability to work with an object after intentionally forgetting how it was built. ​ can interact in complex ways that block understanding ​ More generally, I claim that the opposite of compositionality is emergent effects. The common definition of emergence is a system being ‘more than the sum of its parts’, and so it is easy to see that such a system cannot be understood only in terms of its parts, i.e. it is not compositional. Moreover I claim that non-compositionality is a barrier to scientific understanding, because it breaks the reductionist methodology of always dividing a system into smaller components and translating explanations into lower levels.