nLab proxi-metric space

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 $d(x,y)\leq d(x,z)+d(z,y)$ by the proxi-metric inequality: there exists $C\geq 1$ such that

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

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

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

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

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 $[0,1]$ 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.

Related subjects

global analytic geometry

spectral global analytic geometry