ologs

Ologs, short for ontology logs, are a formal framework for knowledge representation based on category theory. They were developed to provide a precise, compositional way to describe concepts and their relationships, with an emphasis on compatibility with data integration and reasoning. The term is associated with work by David I. Spivak and collaborators.

An olog consists of a directed graph whose nodes are types and whose edges are aspects. Types are typically depicted as boxes, and aspects as arrows between boxes. The construction encourages composition of arrows, so longer chains yield new, derived relations. Equivalently, an olog can be viewed as a small category where objects are types and morphisms are aspects.

Semantics: a model of an olog assigns to each type a set of instances and to each aspect a function between the corresponding sets. The identifications implied by composing arrows are captured by commutative diagrams; a fact in an olog is a statement that a certain diagram commutes. This provides a rigorous, diagrammatic way to express constraints and inferences.

Morphism of ologs preserve structure: a morphism maps types to types and aspects to aspects in a way that respects composition. This enables linking and translating between different ologs, supporting data integration and interoperability. Ologs can be interpreted as database schemas, and functors from an olog to sets provide concrete data interpretations; they can also be realized as relational schemas or RDF graphs.

Example: consider an olog with types Person and City and an aspect livesIn: Person -> City. If another type Country is added with an aspect locatedIn: City -> Country, the composed path Person -> City -> Country yields a derived relation mapping each person to the country where they live. This illustrates how ologs model knowledge compositionally.