Category Theory for Programmers (2014) [pdf]

Conal Elliott's compiling to categories. Categories make a great intermediate representation for DSLs. They are very easy to manipulate/optimize in a simple algebraic manner without issues related to named variables. This proposal showed how a plugin could take ordinary programs written in more user friendly syntax and compile it to the category IR. You can overload ordinary programs to automatically differentiable programs and more. http://conal.net/papers/compiling-to-categories/

Another thing I think is really interesting is calculating programs. It is possible in the right formalism to write down a specification of a program in a simple small form, and use algebraic manipulation to derive an executable form of the program. It feels like doing an integral or something. Check out these books on that topic. https://themattchan.com/docs/algprog.pdf http://www4.di.uminho.pt/~jno/ps/pdbc.pdf

String manipulations of a lifted type in "string space" can be faster and more efficient then actually manipulating it in "type space"

> The "theorems" they then prove do not provide any new insight

Knowing that you can model reaction networks using categorical concepts may not provide new insights IF you have no deep familiarity with those concepts—but if you do, then knowing there is a "symmetric monoidal functor from the category Petri to the category Dynam" is a richly useful piece of information. It's probably more accurate to say that the amount of new insight conferred is proportional to your familiarity with the categorical concepts.

And yes, it's true the concepts are 'just' being translated into a new language—but if this new language is extremely general, applicable to a wide range of domains, then:

1) You can build a set of skills and knowledge related to concepts in this general language, which become automatically applicable to a wide range of domains (when their concepts are imported into the language).

2) It's possible to build 'machinery' (i.e. theorems or software) which deals with / operates on (representations of) concepts from this very general language—so once again, if you can translate some concept from domain X into e.g a symmetric monoidal category, now your general machinery can operate on things from domain X (which it was built with no awareness of).

Additionally, at least in Bradley's paper, the point wasn't to prove new things about e.g. reaction networks; instead it was to illustrate a method. To be more specific she is showing how "functorial semantics" can be applied in this context, using Petri nets as the syntax and dynamical systems as the semantics. From there the categorical classification of these objects gives us guarantees on their behavior under composition (the utility of which I assume is clear).

I'm not sure in the context of reaction networks whether modeling the system with compositionality guarantees and a clean mapping between graph representation and differential equations gives scientists something new or not—but the illustration of this method is done clearly, and the generality of the method is enormous.

And maybe I'm also missing something. I'm not a mathematician, but that's how the situation looks to me.

Look, I completely agree with the general point. I love new ways of thinking about things. But blindly applying extremely complicated tools to domains where their power isn't helpful is not good.

What is happening here is like taking a 2-line program that adds together two numbers and dressing it up into thousands of lines of OOP, until you have AdditionObjectFactoryGetterSetter objects being created by AdditionObjectFactoryGetterSetterFactory objects. Then you can bring in powerful OOP tools, but the only things they help you with are understanding the excess complication you added in. Maybe the sheer scale of that structure is beautiful to someone, but we're discussing whether scientists should bother learning it.

> I'm not sure in the context of reaction networks whether modeling the system with compositionality guarantees and a clean mapping between graph representation and differential equations gives scientists something new or not

And I'm telling you, it really doesn't! There simply is not a rich enough formal structure present in chemical reaction networks for category theory to yield anything useful, like it did in parts of pure math. Adding a complicated structure on top of something simple does nobody any good. All of the "guarantees on compositionality" were not useful, in the sense that they went without saying in freshman-level courses, or worse, were problems solely created by the introduction of category theory.

> the illustration of this method is done clearly, and the generality of the method is enormous.

The fact that the technique is extremely general does not improve things. If a technique makes science harder to do, you can't make it up on volume.