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A network of aligned ontologies is a distributed system, whose components (constituent ontologies) are interacting and interoperating, the result of this interaction being, either the extension of local assertions, which are valid within each individual ontology, to global assertions holding between remote ontology syntactic entities (concepts, individuals) through a network path, or to local assertions holding between local entities of an ontology, but induced by remote ontologies, through a cycle in the network. The mechanism for achieving this interaction is the composition of relations. In this perspective, (a) we introduce the notions of local composable relations, which relate ontology entities belonging to the same ontology, remotely induced composable relations, which relate ontology entities belonging to remote ontologies through a path of ontologies and alignments in the network, and network induced local composable relations, which relate ontology entities belonging to the same ontology, but through a path of ontologies and alignments forming a cycle starting and ending at the same ontology, to characterize the logical consequences extracted from a network of aligned ontologies, and (b) we propose a category-based methodology for detecting semantic inconsistencies in networks of aligned ontologies, which is based on contravariant representable functors and on the definition of two composition operators suitable for propagating local knowledge through the network.

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