On 6 Feb 2009, at 05:52, Gregg Reynolds wrote:
I'm working on a radically different way of looking at IO. Before I post it and make a fool of myself, I'd appreciate a reality check on the following points:
a) Can IO be thought of as a category? I think the answer is yes.
What couldn't? Everything could be thought of as category, linear space, graph or matroid - you'll need some intellectual efforts for that, but it certainly could be done.
b) If it is a category, what are its morphisms? I think the answer is: it has no morphisms.
Oops. Than it's empty. In a category, every object has at least an identity morphism.
c) All categories with no morphisms ("bereft categories"?) are isomorphic (to each other). I think yes.
Yes, all empty categories are isomorphic. No, categories with identity morphisms only are not isomorphic in general. Yes, all categories with no morphisms except identities are equivalent. No, such categories are not very useful and there is no need to apply categorical language to them - thinking in terms of set (class) of objects would be easier.