In this talk, we will try to give a self-contained account on a construction for a directed homology theory based on modules over algebras, linking it to both persistence homology and natural homology, that was originally proposed as a convenient directed homology theory, based on natural systems of modules.

Persistence modules have been introduced originally for topological data analysis, where the data set seen at different «  resolutions »  is organized as a filtration of spaces. This has been further generalized to multidimensional persistence and « generalized » persistence, where a persistence module was defined to be any functor from a partially ordered set, or more generally a preordered set, to an arbitrary category (in general, a category of vector spaces).

Directed topology has its roots in a very different application area, concurrency and distributed systems theory rather than data analysis. Its goal is to study (multi-dimensional) dynamical systems, discrete (precubical sets, for application to concurrency theory) or continuous time (differential inclusions, for e.g. applications in control), that appear in the study of multi-agent systems. In this framework, topological spaces are «  directed », meaning they have «  preferred directions », for instance a cone in the tangent space, if we are considering a manifold, or the canonical local coordinate system in a precubical set. Natural homology, an invariant for directed topology, defines a natural system of modules, a further categorical generalization of (bi)modules, describing the evolution of the standard (simplicial, or singular) homology of certain path spaces, along their endpoints. Indeed, this is, in spirit, similar to persistence homology.

This talk will be concerned with a more « classical » construction of directed homology, mostly for precubical sets here, based on (bi)modules over (path) algebras, making it closer to classical homology with value in modules over rings, and of the techniques introduced for persistence modules. Still, this construction retains the essential information that natural homology is unveiling. Of particular interest will be the role of restriction and extension of scalars functors, that will be central to the discussion of relative homology and Mayer-Vietoris sequences. If time permits as well, we will discuss a Kunneth formula and some « tameness » issues, for dealing with practical calculations.