The 2-category theory of quasi-categories
By Emily Riehl, Dominic R. Verity. 2013
64 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics, Neurology
View Paper: https://www.semanticscholar.org/paper/818f14c5432dac580343cd4373b24770190f7492
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Citations in the corpus
- Homotopy coherent adjunctions and the formal theory of monads (2013, unknown): “The basic category theory of quasi-categories has been developed by Joyal [11,13,12], Lurie [18,19] (under the name ∞-categories), ourselves [23], and others.”
- Fibrations and Yoneda's lemma in an ∞-cosmos (2017, methodology): “2-category theory in an ∞-cosmos Our aim in this section is to demonstrate that all of the basic 2-category theory of quasi-categories developed in sections 3, 4, and 5 of [3] now generalizes to provide corresponding results in the homotopy 2-category of any ∞-cosmos K.”
- ∞-Categories for the Working Mathematician (2018, unknown): “6-8 of [49] • for adjunctions: Proposition 5.”
- Categorical Homotopy Theory (2014, background): “An extended version, which proves that the comma quasi-category of “cones” over a diagram of arbitrary shape is equivalent to the analogous slice category, is given in [RV13a].”