Algebraic & Geometric Topology

Mapping spaces in quasi-categories

Daniel Dugger and David I Spivak

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We apply the Dwyer–Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [Alg. Geom. Topol. 11 (2011) 225–261], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way.

Article information

Algebr. Geom. Topol., Volume 11, Number 1 (2011), 263-325.

Received: 22 December 2009
Revised: 16 August 2010
Accepted: 27 September 2010
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55U40: Topological categories, foundations of homotopy theory
Secondary: 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 18B99: None of the above, but in this section

quasi-category infinity category Dwyer–Kan mapping space simplicial category Joyal model structure homotopy function complex


Dugger, Daniel; Spivak, David I. Mapping spaces in quasi-categories. Algebr. Geom. Topol. 11 (2011), no. 1, 263--325. doi:10.2140/agt.2011.11.263.

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