The 2-category theory of quasi-categories

Abstract

Abstract. In this paper we re-develop the foundations of the category theory of quasicategories (also called ∞-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.

Kan extensions and the calculus of modules for ∞–categories

“…As discussed in §3.5 of [12], a parallel pair of functors a, a ′ : X → f ↓ g are isomorphic over C × B if and only if a and a ′ both enjoy the same defining properties as 1-cells induced by the weak 2-universal property of f ↓ g, i.e., they satisfy…”

“…Previous work [12,15,13,14] shows that the basic theory of (∞, 1)-categoriescategories that are weakly enriched over ∞-groupoids, i.e., topological spaces -can be developed "model independently," at least if one is content to work with one of the better-behaved models: namely, quasi-categories, complete Segal spaces, Segal categories, or naturally marked simplicial sets. More specifically, we show that a large portion of the category theory of quasi-categories-one model of (∞, 1)-categories that has been studied extensively by Joyal, Lurie, and others-can be re-developed from the abstract perspective of the homotopy 2-category of the ∞-cosmos of quasicategories.…”

“…Our work is largely 2-categorical, presented in terms of the ∞-categories, ∞-functors, and ∞-natural transformations that assemble into the homotopy 2-category of some ∞-cosmos, much like ordinary categorical notions can be defined in terms of categories, functors, and natural transformations. References to [12,15,13,14] will have the form I.x.x.x, II.x.x.x, III.x.x.x, and IV.x.x.x respectively. We spare the reader the pain of extensive cross referencing however, by beginning with a comprehensive survey of the necessary background in §2.…”

Left-Induced Model Structures and Diagram Categories

“…Any morphism τ : Φ → Ψ has a canonical presentation as a relative cell complex with cells ℓ r (τ ) ⊗κ r and also a canonical presentation as Postnikov tower with layers m r (τ ) ⋔κ r by [29,Proposition 6.3] and its dual. If τ is a Reedy (acyclic) fibration, then each of its relative matching maps m r (τ ) is an (acyclic) fibration.…”

“…Moreover, if the pair (I, J) defines a cellular presentation for the model category M, then (I ⊗R, J ⊗R) defines a cellular presentation for the Reedy model structure (see [29,Proposition 7.7]). We prove the dual form of this result in Theorem 5.11.…”