Collective adaptive systems (CAS) 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. The distribution of system 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 some logical relations such as being part of the same group. For these systems, specification and verification of spatial properties play a fundamental role to understand their behaviour and to support their design. Recently, different tools and languages have been introduced to specify and verify the properties of space. However, these formalisms are mainly based on graphs. This does not permit considering higher-order relations such as surfaces or volumes. In this work, we propose a spatial logic interpreted on simplicial complexes. These are topological objects able to represent surfaces and volumes efficiently and that generalise graphs with higher-order edges. The expressiveness of the proposed spatial logic is studied in terms of bisimulation and branching bisimulation relations defined over simplicial complexes. Finally, we discuss how the satisfaction of logical formulas can be verified by a correct and complete algorithm.
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