# Category-theoretical Semantics of the Description Logic ALC (extended version)

arXiv: https://arxiv.org/abs/2110.08837 · HTML: https://mlsanity.com/paper/2110.08837/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2021.

## Built on by (most cited first, 4 of 4)

- [Reasoning in the Description Logic ALC under Category Semantics](https://mlsanity.com/paper/2205.04911/) (2022)
- [A New Tractable Description Logic under Categorical Semantics](https://mlsanity.com/paper/2505.08916/) (2025)
- [Lattice-preserving ALC ontology embeddings with saturation](https://mlsanity.com/paper/2305.07163/) (2023)
- [TAPO-Description Logic for Information Behavior: Refined OBoxes, Inference, and Categorical Semantics](https://mlsanity.com/paper/2604.21172/) (2026)

## Abstract (arXiv)

Category theory can be used to state formulas in First-Order Logic without using set membership. Several notable results in logic such as proof of the continuum hypothesis can be elegantly rewritten in category theory. We propose in this paper a reformulation of the usual set-theoretical semantics of the description logic ALC by using categorical language. In this setting, ALC concepts are represented as objects, concept subsumptions as arrows, and memberships as logical quantifiers over objects and arrows of categories. Such a category-theoretical semantics provides a more modular representation of the semantics of ALC and a new way to design algorithms for reasoning.