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-non-linear categories and construct instances of them via Lafont categories and linear categories.

@en

We describe a generic construction of non-wellfounded syntax involving variable binding and its monadic substitution operation.

Our construction of the syntax and its substitution takes place in category theory, notably by using monoidal categories and strong functors between them. A language is specified by a multisorted binding signature, say Σ. First, we provide sufficient criteria for Σ to generate a language of possibly infinite terms, through ω-continuity. Second, we construct a monadic substitution operation for the language generated by Σ. A cornerstone in this construction is a mild generalization of the notion of heterogeneous substitution systems developed by Matthes and Uustalu; such a system encapsulates the necessary corecursion scheme for implementing substitution.

The results are formalized in the Coq proof assistant, through the UniMath library of univalent mathematics. @en

Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. Here, we zoom in on monoidal categories and study them in a univalent setting. Specifically, we show that the bicategory of univalent monoidal categories is univalent. Furthermore, we construct a Rezk completion for monoidal categories: we show how any monoidal category is weakly equivalent to a univalent monoidal category, universally. We have fully formalized these results in UniMath, a library of univalent mathematics in the Coq proof assistant.

@en