What kinds of graphs? There are a number of graphical representations that turn out to be incredibly useful in category theory, but they're far from equivalent. IIRC, https://arxiv.org/abs/1803.05316 introduces some of them.
Going to finish Crime and Punishment, Year of the Monkey (Patti Smith), The Children of Húrin (along with many cross references and letters) (Tolkien), re-reading A Game of Thrones for fun... as for 2020 I might read Slaughterhouse 5 again and work through George Sand.
Technical things I am working through are the:
- https://www.extension.harvard.edu/open-learning-initiative/a... along with the Artin text, writing proofs out by hand and in Lean.
- Seven Sketches in Compositionality: https://arxiv.org/abs/1803.05316
aside from 3Blue1Brown :
I think category theory will give you an unique point of view of math, also basic logic and philosophy are very important for a good and solid foundation on math intuition.
https://arxiv.org/abs/1803.05316 along with their youtube lectures.
very basic but refreshing:
if you are a programmer or developer:
I am currently reading it, it is very nicely written and accessible. Each chapter starts off gently so you can skip to the next whenever you feel lost, only to come back later.
So far: introduction/preorders, monoidal preorders/wiring diagrams, categories/application to database schemas.
For those reading on a ebook, the book sources are available on its arxiv page https://arxiv.org/abs/1803.05316 so you can build a custom pdf for your device.
The mnemonic value of "operators" as reminders of properties such as associativity or commutativity interestingly extends to diagrammatic reasoning. A diagram is really just a generalized expression, and this becomes quite useful when one has to deal with more than one "type" or "domain" of operation, but in a consistent way that preserves the compositionality-like properties OP talks about. A recent book exploring this topic is "Seven Sketches in Compositionality; An Invitation to Applied Category Theory" https://arxiv.org/abs/1803.05316 . (Despite the obvious reference to CT in the title, the work is quite accessible and the math involved is not much more complicated than that found in the linked blogpost. Importantly, and perhaps unlike some people in other programming-language communities, the author does not assume any pre-existing knowledge; the work is rather about using concrete, real-world examples to gently guide the reader's intuition.)
Somewhat less of a "textbook", but good at showing applications of category theory - Fong, Spivak: Seven Sketches in Compositionality https://arxiv.org/abs/1803.05316
(Notably, this includes application to a "graphical" treatment of fairly-elementary linear algebra, which a different HN user is mentioning even as I write this, as a part of math where CT cannot possibly be useful and will only ever confuse students!)