Lawvere-metrics and Banach spaces

[Paul Taylor <pt@cs.man.ac.uk>]

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