We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent bicategories in a modular fashion, we develop displayed bicategories, an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. We demonstrate the applicability of this notion and prove that several bicategories of interest are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Furthermore, we show that every bicategory with univalent hom-categories is weakly equivalent to a univalent bicategory. All of our work is formalized in Coq as part of the UniMath library of univalent mathematics.
Bibliographical noteFunding Information:
We thank Peter LeFanu Lumsdaine and Michael Shulman for helpful discussions on the subject matter, and James Leslie for pointing out several typos in the conference version of this article. We furthermore thank the referees for their careful reading and thoughtful and constructive criticism. Work on this article was supported by a grant from the COST Action EUTypes CA15123. We would like to express our gratitude to all the EUTypes actors for their support. This work was partially funded by EPSRC under agreement number EP/T000252/1. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0363. Ahrens acknowledges the support of the Centre for Advanced Study (CAS) in Oslo, Norway, which funded and hosted the research project Homotopy Type Theory and Univalent Foundations during the 2018/19 academic year. Frumin was supported by the Netherlands Organisation for Scientific Research (NWO/TTW) under the STW project 14319 and VIDI Project No. 016.Vidi.189.046. Maggesi was supported by MIUR and GNSAGA-INdAM. Veltri was supported by the Estonian Research Council grant PSG659 and by the ESF funded Estonian IT Academy research measure (project 2014-2020.4.05.19-0001).
© The Author(s), 2022. Published by Cambridge University Press.
- Bicategory theory
- dependent type theory
- univalent mathematics
ASJC Scopus subject areas
- Mathematics (miscellaneous)
- Computer Science Applications