  The well-known Lawvere category R of extended real positive numbers comes
with a monoidal closed structure where the tensor product is the sum. But R has
another such structure, given by multiplication, which is *-autonomous.
  Normed sets, with a norm in R, inherit thus two symmetric monoidal closed
structures, and categories enriched on one of them have a 'subadditive' or
'submultiplicative' norm, respectively. Typically, the first case occurs when
the norm expresses a cost, the second with Lipschitz norms.
  This paper is a preparation for a sequel, devoted to 'weighted algebraic
topology', an enrichment of directed algebraic topology. The structure of R,
and its extension to the complex projective line, might be a first step in
abstracting a notion of algebra of weights, linked with physical measures.
