Categories , Norms and Weights

The well-known Lawvere category [0,∞] of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But [0,∞] has another such structure, given by multiplication, which is *-autonomous and a CL-algebra (linked with classical linear logic). Normed sets, with a norm in [0,∞], 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 [0,∞], 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.