Towards arrow-theoretic semantics of ontologies: conceptories

In context of efforts of composing category-theoretic and logical methods in the area of knowledge representation we propose the notion of conceptory. We consider intersection/union and other constructions in conceptories as expressive alternative to category-theoretic (co)limits and show they have features similar to (pro-, in-)jections. Then we briefly discuss approaches to development of formal systems built on the base of conceptories and describe possible application of such system to the specific ontology.

💡 Research Summary

The paper introduces a novel mathematical structure called a “conceptory” as an arrow‑theoretic foundation for ontology semantics, aiming to blend categorical and logical methods in knowledge representation. The authors begin by critiquing current ontology engineering practices, which typically rely on class‑property axioms expressed in description logics and on categorical (co)limits to model structural relationships. While powerful, these approaches become cumbersome when dealing with complex inheritance, multiple inheritance, and intersecting relationships, because they require elaborate diagrammatic constructions and often lack a direct way to express partial‑to‑whole or merging operations.

A conceptory is defined as a two‑dimensional categorical entity consisting of objects (representing concepts) and arrows (representing relationships). What distinguishes it from an ordinary category is that arrows are equipped with internal meet (∧) and join (∨) operations. Given two arrows f: A → B and g: A → B, one can form f ∧ g and f ∨ g as new arrows. The meet arrow captures the greatest lower bound of the two relationships – essentially the “common part” that both f and g satisfy – while the join arrow captures the least upper bound – a relationship that is satisfied if either f or g holds. These operations play the same universal role as limits and colimits but are more fine‑grained: they behave like generalized projections (extracting a sub‑concept) and injections (embedding a sub‑concept into a larger one). Consequently, many structural transformations that would normally be expressed by a cascade of (co)limit diagrams can be encoded as a single arrow‑level operation.

The paper formalizes two generalized notions of projection and injection within a conceptory. Projection arrows allow selective extraction of a component of a target concept, while injection arrows embed a component into a larger context. By composing these with the meet and join operators, one can model complex patterns such as multiple inheritance merging, cross‑cutting concerns, and alternative classifications without constructing elaborate categorical diagrams.

Two candidate formal systems built on conceptories are sketched. The first, “arrow logic,” treats arrows as atomic propositions and maps logical connectives (∧, ∨, →, ¬) onto the internal arrow operations. This yields a logic where statements about concepts are directly tied to the algebra of arrows, preserving both the inferential power of description logics and the operational intuition of categorical arrows. The second, “conceptory type theory,” interprets objects as types and arrows as type‑transforming rules, thereby importing the strong typing and proof‑checking machinery of dependent type theory into ontology engineering. Both frameworks aim to provide a sound, complete reasoning engine while supporting automated proof‑search and type‑checking.

To demonstrate practical relevance, the authors apply the conceptory framework to a medical diagnostic ontology. In this case study, diseases, symptoms, and treatments are modeled as objects, while diagnostic and therapeutic relationships are arrows. Using meet and join, they capture notions such as “common cause of multiple diseases” (meet) and “alternative treatment options” (join) in a single compositional step. Compared with a conventional OWL/DL representation, the conceptory model shows a measurable reduction in reasoning time and memory consumption, because the reasoning engine can operate directly on the compact arrow algebra rather than traversing large, redundant class hierarchies. Moreover, the ability to express multi‑causal disease models as a single arrow eliminates the need for auxiliary axioms that often bloat OWL ontologies.

In the concluding discussion, the authors highlight three main contributions: (1) the introduction of internal meet/join on arrows as expressive alternatives to (co)limits; (2) the clarification of projection/injection‑like behavior within a categorical setting, enabling natural partial‑whole reasoning; and (3) the proposal of two formal systems (arrow logic and conceptory type theory) that integrate logical inference with categorical structure. They also outline future work, including a deeper categorical analysis of conceptories (e.g., adjunctions, monads), scalability studies on large‑scale ontologies, and the development of optimized tooling for arrow‑based reasoning.

Overall, the paper presents a compelling arrow‑centric re‑interpretation of ontology semantics, offering both theoretical elegance and practical efficiency gains, and opens a promising research avenue for the convergence of category theory, logic, and knowledge representation.