Weighted algebraic topology, II (Real valued metrics)

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.