Last week I attended a Category Theory meetup, and as usual, it looks like "standard" notations were chosen to be maximally confusing. And then there's implicit lifting everywhere. 1/n
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Category theorists should seriously learn some Lisp. And Conversely, lispers should learn themselves some dependent types.
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Introductions to CT make you believe it's all about preserving the compositional structure of nodes and arrows, but actually, it's about preserving the compositional structure of arbitrarily complex diagrams over increasingly many nested levels of nodes and arrows.
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Category Theory is complex enough that you need all the help you can to understand it, but Category Terrorists invent notation that is maximally confusing with what Mathematicians use, what Computer Scientists use, what Logicians use, and what other Category Terrorists use.
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Some day, I'll understand enough Category Theory to do my own variant with blackjack and hookers—and some coherent notation. And either all the arrows will go from right to left, or function application will be postfix by default.
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The most elementary result of Category Theory, Yoneda's Lemma, requires three levels of arrows, relating internal arrows to external ones that preserve two levels of arrows. Nothing is interesting below three levels: 1. Morphisms, 2. Functors, 3. Natural Transformations.
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There seems to be no convention for diagrams over multiple levels of arrows. Wikipedia's page for the Yoneda Lemma has a semi-nice one with one square inside another, but only position relates two levels, and for more complex diagrams, you'll need at least a third dimension.
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End of conversation
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