IFF Category Theory Ontology

This paper begins the discussion of how the Information Flow Framework can be used to provide a principled foundation for the metalevel (or structural level) of the Standard Upper Ontology (SUO). This SUO structural level can be used as a logical framework for manipulating collections of ontologies in the object level of the SUO or other middle level or domain ontologies. From the Information Flow perspective, the SUO structural level resolves into several metalevel ontologies. This paper discusses a KIF formalization for one of those metalevel categories, the Category Theory Ontology. In particular, it discusses its category and colimit sub-namespaces. The Information Flow Framework The mission of the Information Flow Framework (IFF) is to further the development of the theory of Information Flow, and to apply Information Flow to distributed logic, ontologies, and knowledge representation. IFF provides mechanisms for a principled foundation for an ontological framework – a framework for sharing ontologies, manipulating ontologies as objects, partitioning ontologies, composing ontologies, discussing ontological structure, noting dependencies between ontologies, declaring the use of other ontologies, etc. IFF is primarily based upon the theory of Information Flow initiated by Barwise (Barwise and Seligman 1997), which is centered on the notion of a classification. Information Flow itself based upon the theory of the Chu construction of ∗-autonomous categories (Barr 1996), thus giving it a connection to concurrency and Linear Logic. IFF is secondarily based upon the theory of Formal Concept Analysis initiated by Wille (Ganter & Wille 1999) , which is centered on the notion of a concept lattice. IFF represents metalogic, and as such operates at the structural level of ontologies. In IFF there is a precise boundary between the metalevel and the object level. The structure of IFF is illustrated in Figure 1. This consists of a collection of metalevel ontologies, usually centered on a category-theory category of IFF. ○ At the upper metalevel is the Basic KIF Ontology, whose purpose is to provide an interface between KIF and ontological structure. The Basic KIF Ontology provides an adequate foundation for representing ontologies in general and for defining the other metalevel ontologies of Figure 1 in particular. All ontologies import and use the Basic KIF ontology. ○ At the middle metalevel are three generic ontologies – a Category Theory Ontology (partially presented in this paper) that allows us to make claims about lower metalevel ontologies such as “the Classification Ontology represents a category” or “the classification functor is left adjoint to the IF theory functor” (Kent 2000), a GRAPH Ontology that provides the mathematical context in which the Category Theory Ontology can be defined (as a monoid in a monoidal category), and a SET Ontology which provides foundational axiomatics and terminology for ontologies in the middle or lower metalevel, such as the Category Theory and Classification ontologies. The Category Theory Ontology is a KIF formalism for category theory in one of its normal presentations. Other presentations, such as home-set or arrows-only, may also have value. The Category Theory Ontology provides a framework for reasoning about metalogic, as represented in the lower metalevel ontologies. ○ At the lower metalevel, ontologies are organized in two dimensions, an instantiation-predication dimension and an entity-relation dimension. These two precise mathematical dimensions correspond to the intuitive distinctions of Heraclitus and Peirce. □ In IFF the type-token distinction looms large. In the instantiation-predication dimension are the Classification Ontology that directly represents the type-token distinction, the IF Theory OntolFigure 1: Metalevel Ontologies of IFF Basic KIF