The page is intended to summarize and improve on some ideas in the Discussion at finite category.
One way to obtain a category from a directed graph is to form the free category on that graph. However, not all finite categories can be obtained this way. For example, the “walking commutative square” is not the free category of the walking square graph.
So over at finite category, I was trying to cook up some recipe to obtain any finite category from a directed graph. Then Todd and Mike suggested the use of computads. In fact, Mike said:
It is true that any (finite) category can be obtained as the 1-truncation of the free 2-category on some (finite) 2-computad; in fact this is more or less the definition of a “finitely presented category.”
That is an interesting idea and I definitely want to learn more about computads. However, I am too stubborn to give up on my alternative idea for the time being.
My idea is to specify a directed 2-graph instead of a 2-computad.
Mike Shulman: I don’t think this will work any better than directed 1-graphs. What directed 2-graph are you imagining would generate the walking commutative square?