The notion of participation as a relation between objects, activities, and time has been axiomatized in various ontologies. In this paper, we focus on three of these ontologies – PSL-Core, Gangemi's axioms, and DOLCE. We provide a verification of these participation ontologies by introducing ontologies for new classes of mathematical structures known as incidence bundles and incidence foliations. The new mathematical ontologies serve as reusable ontology design patterns for participation, and also are the basis for mappings between the different participation ontologies. Finally, we illustrate the concept of ontology transfer through the use of these ontology design patterns.