# The Category of Small Categories: a Zoo of Functors.

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 didn't downvote, but I'm not really sure why one would want such a site. – Alex Becker May 9 '14 at 20:51
• I'm not really sure why anyone wouldn't . . . – Shaun May 9 '14 at 20:54
• Regardless of someone's level of experience with categories, I think having a little popup telling you "did you know there is a functor from $\mathsf{Top}_{*}$ to $\mathsf{Grp}$ called the fundamental group functor" is nowhere near as useful as actually knowing the algebraic topology surrounding that construction. In other words, simply looking at a diagram of categories doesn't teach you anything about the actual math. I think I can also safely say that there are no revelations, for the beginner or the expert, obtainable solely by stringing together known constructions between categories. – complexist May 18 '14 at 19:33
• "Regardless of someone's level of experience with Number Theory, I think having a little popup telling you "did you know that there's a sequence of the numbers of trees with $n$ unlabeled nodes?" is nowhere near as useful as actually knowing the Number Theory surrounding that construction. In other words, simply looking at sequences of numbers doesn't teach you anything about the actual math. I think I can also safely say that there are no revelations, for the beginner or the expert, obtainable solely by stringing together known constructions between sequences." – Shaun May 18 '14 at 19:41
• Is possible that a map like this can help someone that is researching in some field and then he can discover that the category he is studyng has links with total different fields. This work can be maybe usefull like a pratical map of mathematics. In my opinion can be usefull and if not really useful, can be really a curious and beautiful thing. I don't know if someone will do (did) this but it need more attention without any doubt:I will add another 50 reps bounty if the user's one will be useless. – MphLee May 18 '14 at 20:23

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:

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.

• +1 I upvoted this as soon as I saw it! (I was on my phone when I commented about an hour ago.) This answer might give anyone serious about it the leg-up they need. Thank you! It's up to @MphLee what he does with his, of course, but I think this has earned a bounty (but not an accept). – Shaun Jun 2 '14 at 9:38
• Corrections: "he or she", "his or hers" - sorry! :) – Shaun Jun 2 '14 at 10:05
• He* was ok @Shaun – MphLee Jun 2 '14 at 11:10
• I seem to be the only interested coder (no one's emailed me yet), so I will work on it when I get time soon. – Shine On You Crazy Diamond Jun 2 '14 at 18:37
• I will just use VS 2010 + WinForms + OpenTK. The cross-platform dev environments are commercial so we'd have to raise money before having it work on anything (eg. Xamarin & Xojo). – Shine On You Crazy Diamond Jun 2 '14 at 19:59