Displayed Monoidal Categories for the Semantics of Linear Logic

Benedikt Ahrens (1, 2) , Ralph Matthes (3, 4) , Niels van der Weide (5) , Kobe Wullaert (1)

1 TU Delft - Delft University of Technology
2 University of Birmingham [Birmingham]
3 IRIT-ACADIE - Assistance à la Certification d’Applications DIstribuées et Embarquées
4 CNRS - Centre National de la Recherche Scientifique
5 Radboud University [Nijmegen]

We present a formalization of different categorical structures used to interpret linear logic. Our formalization takes place in UniMath, a library of univalent mathematics based on the Coq proof assistant. All the categorical structures we formalize are based on monoidal categories. As such, one of our contributions is a practical, usable library of formalized results on monoidal categories. Monoidal categories carry a lot of structure, and instances of monoidal categories are often built from complicated mathematical objects. This can cause challenges of scalability, regarding both the vast amount of data to be managed by the user of the library, as well as the time the proof assistant spends on checking code. To enable scalability, and to avoid duplication of computer code in the formalization, we develop "displayed monoidal categories". These gadgets allow for the modular construction of complicated monoidal categories by building them in layers; we demonstrate their use in many examples. Specifically, we define linear-nonlinear categories and construct instances of them via Lafont categories and linear categories.

Keywords: linear logic; categorical semantics; monoidal categories; Coq; UniMath

Domains: Computer Science [cs]

Benedikt Ahrens, Ralph Matthes, Niels van der Weide, Kobe Wullaert. Displayed Monoidal Categories for the Semantics of Linear Logic. CPP 2024: Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Jan 2024, London, United Kingdom. pp.260-273, ⟨10.1145/3636501.3636956⟩. ⟨hal-04375376⟩