As we have seen, directed algebraic topology studies ‘directed spaces’ with ‘directed algebraic structures’ produced by homotopy or homology functors: on the one hand the fundamental category (and possibly its higher dimensional versions), and on the other preordered homology groups. Its general aim is modelling non-reversible phenomena.

We now sketch an enrichment of this subject: we replace the truthvalued approach of directed algebraic topology (where a path is licit or not) with a measure of costs, taking values in the interval [0,∞] of extended (weakly) positive real numbers. The general aim is, now, measuring the cost of (possibly non-reversible) phenomena.

Weighted algebraic topology will study ‘weighted spaces’, like (generalised) metric spaces, with ‘weighted’ algebraic structures, like the fundamental weighted (or normed) category, defined here, and the weighted homology groups, developed in [G10] for weighted (or normed) cubical sets.

Lawvere's generalised metric spaces, endowed with a possibly nonsymmetric distance taking values in [0,∞] (already considered in Section 1.9.6), are a basic setting where weighted algebraic topology can be developed (see Section 6.1). This approach is based on the standard generalised metric interval δI, with distance δ(x, y) = y–x, if x ≤ y, and δ(x, y) = ∞ otherwise; the resulting cylinder functor I(X) = X ⊗δI has the l1-type metric (Section 6.2). We define the fundamental weighted category wΠ1 (X) of a generalised metric spaces, and begin its study (Sections 6.3 and 6.4).

We work with elementary and extended homotopies, which are given by 1-Lipschitz maps (i.e. weak contractions) and Lipschitz maps, respectively (see Section 6.2).