nLab weighted homotopy theory

References

• Marco Grandis, The fundamental weighted category of a weighted space (From directed to weighted algebraic topology), 2006, arXiv:math/0604506

“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 path-reversion. Thus, ‘weighted algebraic topology’ can be developed as an enriched version of directed algebraic topology, where illicit paths are penalised with an infinite cost, and the licit ones are measured. Its algebraic counterpart will be ‘weighted algebraic structures’, equipped with a sort of directed seminorm.”