The 2-category theory of quasi-categories

TL;DR

This paper develops the foundations of quasi-category theory using 2-category theory, establishing universal properties, limits, and adjunctions with new, formal proofs that generalize to enriched model categories.

Contribution

It introduces a 2-categorical framework for quasi-categories, providing independent proofs and new insights into universal properties and limits in this setting.

Findings

Existence of weak 2-limits in the 2-category of quasi-categories

Encoding universal properties via comma quasi-categories

Universal properties characterized by initial or terminal vertices

Abstract

In this paper we redevelop the foundations of the category theory of quasi-categories (also called infinity-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate form as absolute lifting diagrams in the 2-category, which we show are determined pointwise by the existence of certain initial or terminal vertices, allowing for the easy production of examples. All the quasi-categorical notions introduced here are equivalent to the established ones but our proofs are independent and more "formal". In particular, these results generalise immediately to model categories enriched over quasi-categories.