On $\infty$-cosmoi of bicategories

TL;DR

This paper investigates atypical examples of $ abla$-cosmoi where objects are 2-categories or bicategories, comparing their formal category theory with classical theories to inform future $( abla,2)$-category research.

Contribution

It introduces and analyzes new $ abla$-cosmoi based on bicategories, bridging formal $( abla,1)$-category theory with classical bicategory theory.

Findings

Atypical $ abla$-cosmoi examples are constructed and studied.

Comparison between formal $ abla$-cosmoi theory and classical bicategory theory.

Insights provided for future formal $( abla,2)$-category developments.

Abstract

An $\infty$-cosmos is a setting in which to develop the formal category theory of $(\infty,1)$-categories. In this paper, we explore a few atypical examples of $\infty$-cosmoi whose objects are 2-categories or bicategories rather than $(\infty,1)$-categories and compare the formal category theory that is so-encoded with classical 2-category and bicategory theory. We hope this work will inform future explorations into formal $(\infty,2)$-category theory.