The 2-category theory of quasi-categories
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.
Keywords
Cite
@article{arxiv.1306.5144,
title = {The 2-category theory of quasi-categories},
author = {Emily Riehl and Dominic Verity},
journal= {arXiv preprint arXiv:1306.5144},
year = {2015}
}
Comments
84 pages; v4, final journal version, expository improvements suggested by referee and a handful of new lemmas streamlining the presentation in sections 4-6; an appendix, establishing the equivalence between the join and fat join construction, has been cut from v4 but can be found in v3. 3 slightly expanded section 4, providing background material for the sequel arXiv:1310.8279; v4