Homology theories of directed ‘spaces’ will take values in directed algebraic structures. We will use preordered abelian groups, letting the preorder express most (not all) of the information codified in the original distinguished directions. One could also use abelian monoids, following a procedure developed by A. Patchkoria [P1, P2] for the homology semimodule of a ‘chain complex of semimodules’, but this would give here less information (see Section 2.1.4).

Directed homology of cubical sets, the main subject of this chapter, has interesting features, also related to non-commutative geometry.

Indeed, it may happen that the quotient S/˜ of a topological space has a trivial topology, while the corresponding quotient of its singular cubical set □S keeps relevant topological information, identified by its homology and agreeing with the interpretation of such ‘quotients’ in noncommutative geometry. This relationship, briefly explored here, should be further clarified.

Let us start from the classical results on the homology of an orbit space S/G, for a group G acting properly on a space S; these results can be extended to free actions if we replace S with its singular cubical set and take the quotient cubical set (□S)/G (Corollary 2.4.4 and Theorem 2.4.5). Thus, for the group Gν = Z + ϑZ (ϑ irrational), the orbit space R/Gϑ has a trivial topology (the indiscrete one), but can be replaced with a non-trivial cubical set, X = (□R)/Gϑ, whose homology is the same as the homology of the group Gϑ ≅ Z2, and coincides with the homology of the torus T2 (see (2.76)).