Univalent Double Categories
Niels van der Weide, Nima Rasekh, Benedikt Ahrens, Paige Randall North
TL;DR
The paper develops the univalent bicategory of univalent double categories, proving that the bicategory $\mathsf{DoubleCat}$ is univalent and enabling transport of constructions across equivalences. It formalizes double categories in UniMath/Coq via a layered approach built from 2-sided displayed categories, then shows how horizontal identities, compositions, unitors, and associators can be added modularly while preserving univalence. It further provides concrete univalent double-category examples (e.g., Arr, Span, Lens, structured cospans, Kleisli double categories) and gives criteria for adjoint equivalences and invertible 2-cells within $\mathsf{DoubleCat}$. The work lays a solid foundation for future work on weak/directed variants and monoidal or framed double categories, enabling robust formal reasoning about complex interplays of two morphism types in univalent foundations.
Abstract
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also "morphisms", which capture how different objects interact with each other. Category theory has found many applications in mathematics and in computer science, for example in functional programming. Double categories are a natural generalization of categories which incorporate the data of two separate classes of morphisms, allowing a more nuanced representation of relationships and interactions between objects. Similar to category theory, double categories have been successfully applied to various situations in mathematics and computer science, in which objects naturally exhibit two types of morphisms. Examples include categories themselves, but also lenses, petri nets, and spans. While categories have already been formalized in a variety of proof assistants, double categories have received far less attention. In this paper we remedy this situation by presenting a formalization of double categories via the proof assistant Coq, relying on the Coq UniMath library. As part of this work we present two equivalent formalizations of the definition of a double category, an unfolded explicit definition and a second definition which exhibits excellent formal properties via 2-sided displayed categories. As an application of the formal approach we establish a notion of univalent double category along with a univalence principle: equivalences of univalent double categories coincide with their identities