Gregg Reynolds wrote:
BTW, I'm not talking about Haskell's Functor class, I guess I should
have made that clear. I'm talking about category theory, as the
semantic framework for thinking about Haskell.
Don't forget the part explaining this is just a sketch.
In that case, I even less see why you are not introducing category theory proper. Certainly, if one wants to use a semantic framework for thinking about something, one should use the real thing, not some metaphors.
The idea is that each type (category) is a distinct universe. The essential
point about functors cross boundaries from one category to another.
What are the categories you are talking about here?
Take your pick.
Which part specifically?
Sections 1.5, 1.6, 1.9, 2.1, etc.
Personally, I do not see why one should explain something easy like
functors in terms of something complicated like quantum entanglement.
The metaphor is action-at-a-distance. Quantum entanglement is a vivid way
of conveying it since it is so strange, but true. Obviously one is not
expected to understand quantum entanglement, only the idea of two things
linked "invisibly" across a boundary.
How does the fact that a morphism exists between two objects in some category link these objects together? It doesn't change the objects at all. In your own words: How can action (at-a-distance) be about mathematical values?
Not between two objects in some category; between two objects in different categories. That's the whole point. Functors preserve structure. Action-at-a-distance is a metaphor meant to enliven the concept. You use a map in your home category to map remote objects, by beaming it up through the telefunctor. Your map stays home but is quantum entangled with the remote map. Heh heh. I'm not saying it's for everybody, but I think it's kinda fun.
-g