  Collective Adaptive Systems often consist of many heterogeneous components
typically organised in groups. These entities interact with each other by
adapting their behaviour to pursue individual or collective goals. In these
systems, the distribution of these entities determines a space that can be
either physical or logical. The former is defined in terms of a physical
relation among components. The latter depends on logical relations, such as
being part of the same group. In this context, specification and verification
of spatial properties play a fundamental role in supporting the design of
systems and predicting their behaviour. For this reason, different tools and
techniques have been proposed to specify and verify the properties of space,
mainly described as graphs. Therefore, the approaches generally use model
spatial relations to describe a form of proximity among pairs of entities.
Unfortunately, these graph-based models do not permit considering relations
among more than two entities that may arise when one is interested in
describing aspects of space by involving interactions among groups of entities.
In this work, we propose a spatial logic interpreted on simplicial complexes.
These are topological objects, able to represent surfaces and volumes
efficiently that generalise graphs with higher-order edges. We discuss how the
satisfaction of logical formulas can be verified by a correct and complete
model checking algorithm, which is linear to the dimension of the simplicial
complex and logical formula. The expressiveness of the proposed logic is
studied in terms of the spatial variants of classical bisimulation and
branching bisimulation relations defined over simplicial complexes.
