Jason (jcreed) wrote,

Here is another attempt to take a snapshot of my math-visualization-brain as I read the Eugenia Cheng paper I linked yesterday:

It depicts the associativity property of a monad obtained from composing seven monads. Each side of the equation depicts 42 2-cells; each place two lines of different colors cross is a "distributive law" 2-cell and each place two lines of the same color flow together is the multiplication law of one of the individual monads.

The whole business feels very much like smooshing old ribbon cables flat.

Here's a view of the proof of correctness of associativity, in which it's basically impossible to actually see what is going on, but it looks pretty:

Each plane is an arrow, each line is a 2-cell, and each point is an equation, either a Yang-Baxter identity, or a pentagon identity coming from the definition of distributive law. As you proceed from the top of the shape to the bottom, one association of the monad transforms into the other.

Slightly more perspicuous:
Tags: categories, math
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