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.