I would like to remind the categorical community that Lawvere's paper is now generally available as a TAC Reprints #1. On Tue, 17 Oct 2006, Paul Taylor wrote:
There is a widely cited paper by Bill Lawvere called "Metric spaces, generalised logic and closed catgeories" in which he shows how metric spaces are examples of enriched categories. The enriching structure consists of the nonnegative reals, with "greater than" as the morphisms and addition as the tensor product. Using this one can generalise the notion of metric space by substituting other structures in place of R.
An obvious question is - what happens when we follow through this idea for Banach spaces? What becomes of the $\ell_p$ spaces and of dual spaces? Do families of semi-norms naturally fit into this pattern?
Paul Taylor