**Catarina Faustino**

*CMAT, Univ. Minho*

Directed algebraic topology is a relatively recent area that emerged at the interface between algebraic topology and the field of concurrency theory in computer science. An important line of research in directed algebraic topology is directed homology. In this talk, I will discuss two concepts of directed homology for precubical sets that have been introduced in the literature. Unfortunately, neither of these concepts behaves satisfactorily with respect to the tensor product of precubical sets, which models the parallel composition of independent concurrent systems and is therefore an important categorical construction for precubical sets. In my PhD thesis, I will try to define a notion of directed homology that is compatible with the tensor product.