nLab
proxi-metric space

Contents

Contents

Idea

The category of metric spaces with Lipschitz maps between them is not good enough from a categorical viewpoint: it has finite limits (the max metric on the product) but not all finite colimits: the coproduct doesn’t always exist. To overcome this difficulty, one needs to enlarge this category in various ways.

The first enlargement is obtained by replacing the usual triangular inequality d(x,y)d(x,z)+d(z,y)d(x,y)\leq d(x,z)+d(z,y) by the proxi-metric inequality: there exists C1C\geq 1 such that

d(x,y)Cmax(d(x,z),d(z,y)).d(x,y)\leq C\cdot \max(d(x,z),d(z,y)).

If C=1C=1, we get back the ultrametric inequality, and the usual triangular inequality corresponds to the case C=2C=2.

This gives a category of so-called proxi-metric spaces stable by the + *\mathbb{R}_+^*-flow given by

dd t:=e tlog(d).d\mapsto d^t:=e^{t\log(d)}.

To get enough finite colimits, one needs to restrict to bounded proximetric spaces, and then, one may take an ind-pro or an ind-completion to get a good category of generalized metric spaces, i.e., a convenient setting for the development of a metric stable homotopy theory, based on the use of the interval [0,1][0,1] or a normed version of it.

The aim of this normed/metrized stable homotopy theory is to develop topological cohomological invariants for proxi-normed rings.

global analytic geometry

spectral global analytic geometry

Last revised on February 5, 2016 at 05:31:58. See the history of this page for a list of all contributions to it.