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I've seen 2-categories diagrams, where there are 2-cells and 1-cells morphisms, but I do not understand how to read them.

Apparently it seems to be some kind of rule that says that you have to follow the paths (as in a horizontal composition) and then vertically compose all of them? Is there any place when I can read how to read them?

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1 Answer 1

The rule is basically the same you learned when you saw your first commutative diagram of 1-cells. 2-cells go between 1-cells and are composable whenever the domain of the first equals the codomain of the second. Now you have to read accordingly to build other diagrams you ask to commute, using one of the various representation of higher categories as suitable presheaves of "abstract shapes"

The only complication is that higher dimensional cells can be composed in more than one way (since they expand in more than one spatial direction)... The other complication is that higher cells are the basic tool to express the "conditional commutativity" that holds in higher categories: geometric intuition can be used in low dimension, but I dare anybody to properly visualize commutativity conditions in dimension higher than 3 :)

You are probably in need of this geometric intuition. I suggest you to have a look at Lauda/Cheng book on higher category theory: you never tasted real categorical pleasure if you never built your own associahedron :)

I think that the double-categorical approach which represents 2-cells $\alpha$ as squares like $$ \begin{array}{ccc} x &\xrightarrow{f}& y \\ \downarrow&\alpha &\downarrow\\ z &\xrightarrow[g]{}& w \end{array} $$ is better suited than the "globular" approach: as an example, I find horizontal composition far more intuitive from the visual point of view.

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