The IFF Foundation (meta) Ontology

The IFF Foundation Ontology comprises the terminology and axiomatization for all the ontologies in the IFF top and upper metalevels. The IFF Foundation Ontology is partitioned into the IFF Top Core(Basic KIF) (meta) Ontology, the IFF Upper Core(meta) Ontology, the IFF UpperRelation-Order (meta) Ontology, the IFF UpperClassification (meta) Ontology and the IFF Upper CategoryTheory (meta) Ontology. The terminology in the IFF Foundation Ontology is conceptual - terms represent concepts (Figure 1). There are ~50 (non-parameterized) concepts in the core, relation-order and graph parts of the IFF Foundation Ontology. These are connected by ~850 functions representing, either attributes of these concepts, other parameterized concepts, transformations between concepts, etc.

There are 1058 concepts or non-identical terms (synonyms are equated) in the IFF Foundation Ontology. There are 63 concepts in the IFF Top Core (Basic KIF) Ontology. The 277 non-identical terms in the IFF Upper Core Ontology are partitioned into:

• 49 terms for classes and functions (SET);
• 131 terms for limits in the quasi-category of classes and functions (SET.LIM); and
• 97 terms for colimits in the quasi-category of classes and functions (SET.COL).

The 130 non-identical terms in the IFF Upper Relation-Order Ontology are partitioned into:

• 46 terms in the namespace (REL) for large relations; and
• 84 terms in the namespace (ORD) for large orders and their various morphisms.

The 286 non-identical terms in the IFF Upper Classification Ontology are partitioned into

• 32 terms for large concept lattices (CL, CL.MOR),
• 41 terms for large complete lattices (LAT, LAT.ADJ, LAT.MOR), and
• 213 terms for large classifications (CLS, CLS.CL, CLS.FIB, CLS.COLL, CLS.INFO, CLS.REL, CLS.BND, CLS.BNDPR, CLS.COL, CLS.COL.COPRD, CLS.COL.COINV, CLS.COL.COEQ, CLS.COL.PSH).

The 302 non-identical terms in the IFF Upper Category Theory Ontology are partitioned into:

• 78 terms for large graphs and their morphisms (GPH);
• 43 terms for categories (CAT);
• 31 terms for functors (FUNC);
• 15 terms for natural transformations (NAT);
• 21 terms for adjunctions (ADJ, ADJ.MOR);
• 37 for monads (MND, MND.EM, MND.KLI); and
• 77 terms for colimits (COL, COL.INIT, COL.COPRD2, COL.COEQ, COL.PSH).

In general in the IFF, terms (concepts) can be either abstract or concrete. Abstract terms are axiomatized by using concepts (hence, terms) from the ambient level upwards, whereas concrete terms also make use of terms below the ambient level. For example, composition in categories is abstract, whereas smallness prop-erty of categories is concrete.