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
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The 2-category theory of quasi-categories. / Riehl, Emily; Verity, Dominic.
In: Advances in Mathematics, Vol. 280, 06.08.2015, p. 549-642.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 -