preprint: Categories, norms and weights

preprint: Categories, norms and weights

[Marco Grandis]

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.