Category theory has a high ratio of definitions to theorems, and it can seem like every definition depends on an infinite regress of definitions. But if you graph the definition dependencies, the graph is wide rather than deep.
I picked out 65 concepts and created a visualization using GraphViz. No definition is more than six degrees away from first principles, and most are closer. The graph is pretty big, so here’s a thumbnail.
The full graph is available here.
It’s possible to define concepts in different ways and obtain a different graph. I used definitions from here.
Here is a list of other math diagrams I’ve created.
Related: Applied category theory
2 thoughts on “Visualizing category concept dependencies”
If you follow Mac Lane’s dictum, ‘All concepts are Kan extensions’, I bet it would be even shallower… :-)
Neat idea. I found some of those dependencies a little surprising – not that there is a standard treatment for CT, and as wonderful as “The Joy of Cats” might be, I wonder how close it is to what Category Theorists would consider a standard presentation.
The notion of a definition tree seems pretty interesting. Naively, the depth of a definition in the tree should correspond to its conceptual depth. The “stability” of the depths and connections between concepts across various treatments shows how standardized the subject has become. Resemblances between trees can show where various treatments have borrowed from each other, or where legitimate differences exist.