This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field.
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Yeah it's not ideal. Also with the current preference for category theory it would probably make more sense to not phrase E as a subset of NxN but rather to define two functions pre: E -> N and post: E -> N.
You could then choose to add labels as either an additional function label: E -> L, or as several graphs layered on top of one another (especially helpful when you view each graph as an arrow N -> N, which in turn makes more sense when you have several different classes of nodes).
If anyone's interested I'm loosely basing this description on the description of petri nets as outlined in Tai-Danae Bradley's interesting (and readable) Applied Category Theory: https://arxiv.org/pdf/1809.05923.pdf
This beautiful booklet my Tai-Danae Bradley also contains hand-drawn diagrams:
https://arxiv.org/pdf/1809.05923.pdf
It is set using the Tufte LaTeX template:
https://twitter.com/math3ma/status/1042569522056716289
https://www.overleaf.com/gallery/tagged/tufte
I'm no expert, so the best I can do is point you to the research community gathering around applied category theory.
* A conference series + workshop: https://www.appliedcategorytheory.org/
* A journal: https://compositionality-journal.org/
Check out some of the people involved in organizing these events. Names I've followed include John Baez, Pawel Sobocinski, and Tai-Danae Bradley (all of whom are amazing educators; check out their work!).
Tai-Danae Bradley wrote a pamphlet on applied category theory which is very approachable: https://arxiv.org/pdf/1809.05923.pdf . Also check out Jules Hedges' thoughts on the importance of compositionality: https://julesh.com/2017/04/22/on-compositionality/ .
The clearest and most concise treatment I've come across on how Category Theory is used is Tai-Danae Bradley's "What is Applied Category Theory": https://arxiv.org/abs/1809.05923 (It's a 50 page PDF)
Unfortunately the examples aren't within CS, but at least for me personally, seeing it clearly applied in any context was the most important thing. It may be a good pre-req for looking at (or conjuring up your own) more detailed applications within CS specifically.
Also, if you like that, Tai-Danae Bradley has more writing at: https://www.math3ma.com/
For those wanting these ideas described in a more mathematical "language" [0-1] are a good start. If you are interested in studying objects and how they compose, you are probably interested in category theory.
[0] https://golem.ph.utexas.edu/category/2018/02/linguistics_usi...
https://arxiv.org/pdf/1809.05923.pdf
[1] https://arxiv.org/pdf/1809.05923.pdf
Tai-Danae Bradley has a very nice write-up called "What is applied category theory?" -- https://arxiv.org/abs/1809.05923
Among the more readable category theory resources I'm currently reading.
In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.
Artificial intelligence (AI) researchers have been developing and refining large language models (LLMs) that exhibit remarkable capabilities across a variety of domains and tasks, challenging our understanding of learning and cognition.