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Categories, norms and weights
#
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