  Higher-dimensional automata (HDAs) are models of non-interleaving concurrency
for analyzing concurrent systems. There is a rich literature that deals with
bisimulations for concurrent systems, and some of them have been extended to
HDAs. However, no logical characterizations of these relations are currently
available for HDAs.
  In this work, we address this gap by introducing Ipomset modal logic, a
Hennessy-Milner type logic over HDAs, and show that it characterizes
Path-bisimulation, a variant of the standard ST-bisimulation. We also define a
notion of Cell-bisimulation, using the open-maps framework of Joyal, Nielsen,
and Winskel, and establish the relationship between these bisimulations (and
also their "strong" variants, which take restrictions into account). In our
work, we rely on the new categorical definition of HDAs as presheaves over
concurrency lists and on track objects.
