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Categories, norms and weights

Abstract:
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 struc- tures, 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 pro jective line, might be a first step in abstracting a notion of algebra of weights, linked with physical measures.

Keywords:
Monoidal categories, closed categories, enriched categories.

MSC:
18

Pubblicato su: Journal of Homotopy and Related Structures Vol. 2 (2007) N. 9 Pag. 171-186