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 by the proxi-metric inequality: there exists such that
If , we get back the ultrametric inequality, and the usual triangular inequality corresponds to the case .
This gives a category of so-called proxi-metric spaces stable by the -flow given by
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 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.
Last revised on February 5, 2016 at 10:31:58. See the history of this page for a list of all contributions to it.