Quasi-categories vs Segal spaces

By André Joyal, M. Tierney. 2006

278 citations recorded by the source corpus.

We show that complete Segal spaces and Segal categories are Quillen equivalent to quasi-categories.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cfc84f956bce5e019bc983c94200844f36a660cf

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The theory and practice of Reedy categories (2013, unknown): “ass of coﬁbrations and a class of ﬁbrations deﬁne a model structure if andonlyifthereareapairofweakfactorisationsystems givenbythetrivialcoﬁbrations andﬁbrations andthecoﬁbrations andtrivialﬁbrations [7]. Twoclassesof mapsforma weakfactorisationsystemiftheysatisfytheliftingandfactorisationproperties”
The 2-category theory of quasi-categories (2013, unknown): “The canonical example [11,21] is certainly included under this heading but we have in mind more general “Rezk spaces” as well.”
∞-Categories for the Working Mathematician (2018, background): “(ii) There is also a cosmological biequivalence 𝒬𝒞𝑎𝑡 ⥲ 𝒞𝒮𝒮 defined by Joyal and Tierney [26].”
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.”
A universal characterization of higher algebraic K-theory (2010, background): “ ∆1 and presheaves of ∞-categories on ∆1 [52, 3.2.0.1]. In order to understand the condition of being fully faithful, we write Cat∞ as an accessible localization Cat∞ ⊆ Pre(N(∆)) of simplicial spaces [47]. A functor is fully faithful when the corresponding map of (local) simplicial spaces is fully f”
Dendroidal sets as models for homotopy operads (2009, background): “The equivalence of these approaches is extensively discussed in Lurie’s book [Lur06]; see also [Ber07, JT07].”
On an extension of the notion of Reedy category (2008, background): “[6,25,27,35]), there is a localization of the model category of dendroidal spaces which is Quillen equivalent to the model category of quasi-operads introduced in [11].”
A cartesian presentation of weak n–categories (2009, background): “This is recognized as a suitable model for (∞, 1)-categories, due to work of Bergner [BerJ07b] and Joyal-Tierney [JT07].”
Categorical Homotopy Theory (2014, background): “The following definition, perhaps first due to [JT07], a source of several useful facts about model categories, is equivalent to the usual one.”
Higher Segal Spaces (2012, background): “6 in [JT07], we conclude that Spanpt(C) is an ∞-category.”
A type theory for synthetic ∞-categories (2017, background): “The homotopy theoretic analysis of Joyal and Tierney [11] suggests that it is productive to think of bisimplicial sets as having a “spatial” direction and “categorical” direction; simplicial sets can then be embedded in the categorical direction as discrete simplicial spaces or in the spatial direct”
Fibrations and Yoneda's lemma in an ∞-cosmos (2017, unknown): “We adopt notation from [15], and in particular, make use of the “external product” bifunctor − − : sSet× sSet → ssSet.”