On Mon, Feb 2, 2009 at 3:27 PM, David Menendez
Does that help at all?
I think it does. But ... it gives me crazy ideas. Like: a functor is a kind of magic non-computing function! That's why they didn't call it a function? We know it maps A to FA, but we don't know how (maybe we don't care): there's no algorithm, just a functorific magic carpet that transports us across the border to FA. We couldn't compute FA even if we wanted to - different categories are like alternate universes, it would be like producing a widget in an alternate physical universe, we have no way of even thinking of how to do that. Ditto for data constructors as natural transformations: they don't compute, they just do magic. They're the CT surgeon's devious way of working on the guts of a categorical object without getting his hands dirty with mundane functions - getting from value A to (an?) image value of A under the functor F, which we cannot do directly by algorithm or computation. We do not - can not - have an actual function that computes A's image. We have to work indirectly, using non-computing magic carpets - first Id takes us to Id A, then we follow the nat trans to FA. Ok, that probably triggers the gag reflex for a real mathematician, but it sure sounds like a good story to me (remember, I'm taking notes for a guide/tutorial for newbies that may or may not ever get written.) Thanks very much for the help, -gregg