The following preprint is available: Marco Grandis Categories, norms and weights Dip. Mat. Univ. Genova, Preprint 538 (2006), 14 p. http://www.dima.unige.it/~grandis/wCat.pdf http://www.dima.unige.it/~grandis/wCat.ps Abstract. 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 and a CL-algebra (linked with classical linear logic). 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.