On my interactive category theory demonstrations page, I ask for suggestions about how to present categorical concepts graphically. I'd like suggestions that present the concepts in an "intuitive" or "obvious" way, whatever that means to you. In other words, I'd like to ask you — if it's not too embarrassing — to make your private metaphors public. Perhaps I can implement some of them in graphics. But I would also like to try setting up a catalogue of such metaphors, perhaps as a wiki.
New Omar Ahmad recommended pictures from Eugenia Cheng's YouTube talk on General limits and colimits 2, as helpful in ways of visualising how pullbacks, products, and equalizers are all limits of diagrams.
Here are a few quotes around this. Most appositely, perhaps, Dan Piponi's A Monadic Example, from the n-Category Café blog:
When programming in the language Haskell beginners almost immediately hit the notion of category theoretical monads. Even the simplest "Hello, World!" program requires the use of monads. What I think is interesting about this is that Haskell beginners, many of whom have very few abstract mathematics skills, let alone category theory, suddenly have to leapfrog into completely foreign territory. They can try to read the formal definition of categories, functors and monads, and may even be able to carry out basic proofs with them. But the theory seems completely uninteresting to the beginner, especially when all you want to do is print "Hello, World!".
So we have a situation where people have been forced to find ways to make monads accessible and interesting, and now there's a whole industry of people writing 'narratives' with florid metaphors in an attempt to get the meaning over clearly. Various writers have used metaphors like containers, spacesuits(!), storage for nuclear waste(!!), types of piping and nesting, Hotel California (you read that correctly someone thinks monads are like the Hotel California!), computations, and 'unsafe' functions. It's incredible how creative technical writers can become when there's a definite need. It's also interesting to see people make their private metaphors public like this. I'm sure that all mathematicians have lots of bizarre and interesting metaphors for mathematical concepts that they wouldn't normally share with other people.
The objective of such metaphors should be to make the concepts obvious. Here's a remark by John Baez in another n-Category Cafe post, on Re: Why Mathematics Is Boring:
I used to try to prove 'tricky' or 'difficult' things, and it never worked. I kept finding mistakes in my proofs. Eventually I gave up and decided to only prove stuff that was completely obvious to me. The challenge then became making lots of things completely obvious!
Or think of Ronald Brown and Timothy Porter's comment on notation in Section 3 of Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience:
Indeed the number of conventions you need to understand equation (3) make it seem barbaric compared with the picture (4).
Or (thanks to Steve Vickers for this quote) something that Grothendieck said, described in Allyn Jackson's Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck:
One thing Grothendieck said was that one should never try to prove anything that is not almost obvious. This does not mean that one should not be ambitious in choosing things to work on. Rather, "if you don't see that what you are working on is almost obvious, then you are not ready to work on that yet," explained Arthur Ogus of the University of California at Berkeley. "Prepare the way. And that was his approach to mathematics, that everything should be so natural that it just seems completely straightforward." Many mathematicians will choose a well-formulated problem and knock away at it, an approach that Grothendieck disliked. In a well-known passage of Récoltes et Semailles, he describes this approach as being comparable to cracking a nut with a hammer and chisel. What he prefers to do is to soften the shell slowly in water, or to leave it in the sun and the rain, and wait for the right moment when the nut opens naturally (pages 552-553). "So a lot of what Grothendieck did looks like the natural landscape of things, because it looks like it grew, as if on its own," Ogus noted.
15th January 2009