On Wed, Jun 2, 2010 at 9:27 PM, Bill Gosper <billgosper@gmail.com> wrote:
My tutor Julian points out an inconvenient fact: R x (R x R) = {(x,(y,z))} and (R x R) x R = {((x,y),z)} and neither = Euclidean 3Space {(x,y,z)} = R x R x R . What is the approved way to sweep this under the rug? --rwg
You learn a little category theory! Products are unique only up to specified isomorphism. A product of objects A and B is - an object S together with - morphisms i:S->A, j:S->B such that - for any object T and - morphisms p:T->A and q:T->B, there's a unique morphism - h:T->S such that - p = i o h and - q = j o h. We typically use "A x B" instead of S, and "i x j" instead of h. This generalizes easily to n objects. If our objects are sets and our morphisms are functions, then we get the cartesian product. If our objects are sets and there's a unique morphism from X to Y if X is a subset of Y, then the product is the intersection of sets. If our objects are natural numbers and there's a unique morphism from X to Y if X <= Y, then the product is min(X, Y). If our objects are natural numbers and there's a unique morphism from X to Y if X | Y, then the product is gcd(X,Y). -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com