Wouldn't it be great if there was some website or something that visualized (some small portion of) the category of small categories(*)?
Imagine you click on some categories from a list, say, then "pop!" - they appear as dots in a digraph (that's not necessarily a commutative diagram) you can reshape and move around, where each arrow is a (standard) functor. If you click on an arrow, it gives you some details about the functor.
Does such a thing exist?
If not, someone please make it! I haven't found anything like it so far and I don't understand why. It could be quite appealing if done properly. It might even be useful, like the OEIS, as something to explore the subject with :)
Something like this would be spectacular! :D
(*) . . . or the quasicategory of all categories.
I've had fantasies of a similar general "math explorer" project that mathematicians can go to to look up theorems, definitions, vote on elegant proofs, etc. It would also have a 3D viewer to view all related theorems in an area and and you can see the areas that "need work" (for instance). I also want it to implement a fuzzy parser that understands through typos and even word substitution and can be updated with new input-to-output mappings very easily (ie. you don't have to write a grammar of valid input, but merely provide an example and what you want the output to be, or map it to another valid fuzzy input).
As you can see your project if left open to innovative features, which I think it should, will have a few non-trivial computer science & engineering challenges.
These links should get us started if anyone else is serious:
Project:
GitHub Repo For Hosting Code
Project Brainstorm Concept Map
Algorithms:
Libraries & Tools:
Related:
First thing we need to decide is a primary development language (for the explorer gui), or even if we're going to make a standalone app at all.
Please email me for editing privelages on the concept map:
enjoysmath at gmail.
Let me know if you'd like to contribute and need help setting up any of the tools.