Title
Univalent Double Categories
Author
van der Weide, N.J. (Radboud Universiteit Nijmegen)
Rasekh, Nima (Max Planck Institute for Mathematics)
Ahrens, B.P. (TU Delft Programming Languages; University of Birmingham)
North, P.R. (Universiteit Utrecht)
Contributor
Timany, Amin (editor)
Traytel, Dmitriy (editor)
Pientka, Brigitte (editor)
Blazy, Sandrine (editor)
Date
2024
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.
Subject
category theory
double categories
formalization of mathematics
univalent foundations
To reference this document use:
http://resolver.tudelft.nl/uuid:6edbc2e4-1cc2-458c-a273-5559bcf009f5
DOI
https://doi.org/10.1145/3636501.3636955
Publisher
Association for Computing Machinery (ACM), New York, NY, USA
ISBN
979-8-4007-0488-8
Source
CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024
Event
13th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2024, in affiliation with the annual Symposium on Principles of Programming, Languages, ,POPL 2024, 2024-01-15 → 2024-01-16, London, United Kingdom
Series
CPP 2024 - Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, Co-located with: POPL 2024
Part of collection
Institutional Repository
Document type
conference paper
Rights
© 2024 N.J. van der Weide, Nima Rasekh, B.P. Ahrens, P.R. North