Hi, Ellis, Very ambitious questions you're asking here. I'm not sure there's much in the way of clear-cut answers, though. To begin with, I'm not nearly as sure as you are that
a great deal more is known about mathematical rigor than about mathematical intuition.
Or perhaps there is -- what particularly did you have in mind that "is known about mathematical rigor"?
My overall question is whether there really are different kinds of intuition depending on the research discipline.
Well, "depending on the research discipline"? Who can tell? But "whether there really are different kinds of intuition"? Almost surely yes. Both within and across disciplines. Just as people have different complexions, they'll have "different kinds of intuition".
... In particular, is there some kind of kinetic intuition specific to category theory that crucially involves visualization of time-varying diagrams? Do conjectured adjoint functors arise from distinct algebraic, or geometric, or logical intuitions? Do categorists deploy special methods to access their intuition, or do intuitions just happen to those with a knack for category theory?
Intuitions, I'd say, "just happen to those with a knack for" intuitions. And no, I'm not just being difficult. I believe you're suffering from a sort of Aristotelian disease, common to many even two millenia after Aristotle, that believes any sufficiently well-formed and narrowly specific question must have a clear yes-no, or "this-or-that", answer. How can someone like you, who's really smart enough not to fall for that, not be smart enough not to fall for that? -- because Aristotelean "yes/no" black-or-white logic doesn't generally apply to the real world, with all its gradations and shades of gray :-) .
Does categorical intuition just develop with experience, or is there a specialized training to enhance it?
Why not both? or neither? or other?
... Is categorical intuition any different from mathematical intuition in general?
At the risk of sounding more Zen than I intend, I'd answer "Of course it's different. And yet it's not different. In fact it's both different and not different. As well, it's neither different, nor not different." "Do I contradict myself? Very well, then, I contradict myself." :-) .
Ellis D. Cooper
Cheers, -- Fred (with a tip o' the ol' hat to Carl Sandburg) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]