Outdated Archival Version

These pages are not updated anymore. They reflect the state of 6 June 2010. For the current production of this journal, please refer to http://www.springer.com/mathematics/geometry/journal/40062.

Categories, norms and weights

Marco Grandis

The well-known Lawvere category $ \zety $ of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But $ \zety $ has another such structure, given by multiplication, which is *-autonomous and a CL-algebra (linked with classical linear logic). {\it Normed sets}, with a norm in $ \zety$, 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 {\it weighted} algebraic topology, an enrichment of {\it directed} algebraic topology. The structure of $ \zety$, and its extension to the complex projective line, might be a first step in abstracting a notion of {\it algebra of weights}, linked with physical measures.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 171-186