In the IFF, we can mathematically define the notions of monocosmic and polycosmic modularized object-level ontologies; and from the metatheoretic standpoint the notion of a polycosmic diagram of theories is very natural.

In the IFF, a library of modules is conceptually situated within the context of a lattice of theories (generalization/specialization hierarchy) and its correlated structure known as the truth concept lattice. In the IFF, an “unpopulated monolithic object-level ontology” is represented as an IFF theory. In the IFF, a library of modules is regarded as “an unpopulated modularized object-level ontology”, and represented by a diagram of IFF theories and theory morphisms.

It is important to note that the theories within a diagram of theories do not necessarily have the same underlying language. To semantically compare these theories and to conceptually situate them within the context of a lattice of theories, we move them to the lattice of theories over the colimit of the base language. For any diagram of theories, there is a direct information flow diagram in the lattice of theories over the colimit language. This is the direct flow along the colimit injections. The direct information flow is a diagram of theories that all have the same underlying base language, and hence can be semantically compared.

A diagram of theories is monocosmic when there is a model with underlying type language the colimit language, which satisfies all the theories in the direct information flow. A diagram of theories polycosmic when it is not monocosmic; that is, when there is no model which satisfies all the theories in the direct information flow; that is, when there are two mutually inconsistent theories in the direct information flow. In the IFF-OBJ, there are some extreme polycosmic diagrams of theories, where any two theories are either equivalent or mutually inconsistent. Each of these theories lies at the lowest level in the lattice of theories strictly above the bottom inconsistent theory containing all expressions.