### Abstract

In this paper we re-develop the foundations of the category theory of quasi-categories (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.

Language | English |
---|---|

Pages | 549-642 |

Number of pages | 94 |

Journal | Advances in Mathematics |

Volume | 280 |

DOIs | |

Publication status | Published - 6 Aug 2015 |

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### Keywords

- Quasi-categories
- 2-Category theory
- Formal category theory

### Cite this

*Advances in Mathematics*,

*280*, 549-642. https://doi.org/10.1016/j.aim.2015.04.021

}

*Advances in Mathematics*, vol. 280, pp. 549-642. https://doi.org/10.1016/j.aim.2015.04.021

**The 2-category theory of quasi-categories.** / Riehl, Emily; Verity, Dominic.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - The 2-category theory of quasi-categories

AU - Riehl, Emily

AU - Verity, Dominic

PY - 2015/8/6

Y1 - 2015/8/6

N2 - In this paper we re-develop the foundations of the category theory of quasi-categories (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.

AB - In this paper we re-develop the foundations of the category theory of quasi-categories (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.

KW - Quasi-categories

KW - 2-Category theory

KW - Formal category theory

UR - http://www.scopus.com/inward/record.url?scp=84929193052&partnerID=8YFLogxK

UR - http://purl.org/au-research/grants/arc/DP1094883

U2 - 10.1016/j.aim.2015.04.021

DO - 10.1016/j.aim.2015.04.021

M3 - Article

VL - 280

SP - 549

EP - 642

JO - Advances in Mathematics

T2 - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -