The fundamental weighted category of a weighted space: From directed to weighted algebraic topology

Abstract

We want to investigate ‘spaces’ where paths have a ‘weight’, or ‘cost’, expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to pathreversion. Thus, ‘weighted algebraic topology’ can be developed as an enriched version of directed algebraic topology, where illegal paths are penalised with an infinite cost, and the legal ones are measured. Its algebraic counterpart will be ‘weighted algebraic structures’, equipped with a sort of directed seminorm. In the fundamental weighted category of a generalised metric space, introduced here, each homotopy class of paths has a weight (or seminorm), which is subadditive with respect to composition. We also study a more general setting, spaces with weighted paths, which has finer quotients and strong links with noncommutative geometry. Weighted homology of weighted cubical sets has already been developed in a previous work, with similar results.