<div class="gmail_quote">On Mon, Feb 2, 2009 at 3:27 PM, David Menendez <span dir="ltr"><<a href="mailto:dave@zednenem.com">dave@zednenem.com</a>></span> wrote:<br><div> </div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
Does that help at all?<br>
</blockquote><div><br>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.<br>
<br>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.<br>
<br>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.)<br>
<br>Thanks very much for the help,<br><br>-gregg<br></div></div><br>