Weighted algebraic topology, II (Real valued metrics)

TL;DR

This paper develops the theory of real-valued metrics within algebraic topology, extending Lawvere's metric spaces to measure processes like energy or profit, and explores their categorical structures.

Contribution

It introduces real metrics as enriched categories with values in the extended real line, expanding the framework of weighted algebraic topology.

Findings

Real metrics form a symmetric monoidal closed category.

Linear real metrics derive from potential functions.

Part of a series on weighted algebraic topology with future extensions.

Abstract

Extending the `metric spaces' of Lawvere, we study `real metrics', with values in the extended real line. Formally, this ordered set is a symmetric monoidal closed category, and our structures are enriched categories on the latter. Concretely, the present goal is measuring `profits' and `losses' of a process, in any sense - possibly related to energy, or a variable in any science. In particular, linear real metrics derive from a potential function. This article is Part II in a series devoted to `weighted algebraic topology' - an enriched version of directed algebraic topology, where paths are measured. Part III will introduce a finer framework, more adequate to `quotient spaces' (as the spheres) and better related to topology.