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Publication . Article . Preprint . Conference object . 2021

Bicategories in univalent foundations

Benedikt Ahrens; Dan Frumin; Marco Maggesi; Niccolò Veltri; Niels van der Weide;
Open Access
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.
v1: 16 pages; v2: Veltri added as coauthor, extended version, 32 pages, list of changes given in Section "Publication history"; v3: final journal version to be published in Mathematical Structures in Computer Science; v4: fixed some typos that remain in the MSCS version

Bicategory theory, univalent mathematics, dependent type theory, Coq, Computer Science Applications, Mathematics (miscellaneous), Computer Science, 000 Computer science, knowledge, general works, bicategory theory, univalent mathematics, dependent type theory, Coq, Mathematics - Category Theory, Computer Science - Logic in Computer Science, Category Theory (math.CT), Logic in Computer Science (cs.LO), FOS: Mathematics, FOS: Computer and information sciences