A normal category is built on the simple notion of a graph defined with two functions and two types
$s, t: Arrows \to Vertices $
Add to that a notion of a path and equations between paths and one gets the notion of a category. Indeed it is well known that every Graph gives rise the Free Category of that graph.
But what about 2-categories? Those have morphisms between morphisms, so it seemed to me that a simple graph as above won't do. (Or I could not understand the magic reasoning that would get me there)
So I asked what structure would play the same role on Twitter and Eugenia Cheng answered that
Yes, you need 2-cells —s,t—> arrows —s,t—> vertices. ie a 2-graph.
So that's a good concise answer for Twitter, and it gave me confidence that I had asked a good question. But I could not find much about 2-graphs online. It is clearly the right answer as the ncatlab page on strict 2-categories mentions them, but then links to a page with a definition of globular sets which is completely opaque to me.
Can anyone perhaps develop the point made by Eugenia, and perhaps point to a resource that describes 2-graphs in more detail? I would have expected some document to show how one builds paths on 2-graphs to get a 2-category, the way it is done with simple categories.
Best Answer
Since they don't have composition, $2$-graphs are much easier to describe than $2$-categories. You have a set of $0$-morphisms (objects), for any two objects, you have a set of 1-morphisms between them, and for any two parallel $1$-morphisms, you have a set of 2-morphisms between them.
That's it. As you can imagine, this is pretty easy to generalize to $n$-graphs or $\infty$-graphs, which are more commonly called globular sets. However, in that context, you often consider all of the $k$-morphisms as being in one set. That means that you need some extra data about the source and target of each. This turns out to be easier to work with, since it means that a globular set is just a functor from a particular category (whose objects are natural numbers) to the category of sets.