Volume 20, issue 6 (2020)

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A relative $2$–nerve

Fernando Abellán García, Tobias Dyckerhoff and Walker H Stern

Algebraic & Geometric Topology 20 (2020) 3147–3182
DOI: 10.2140/agt.2020.20.3147

We introduce a 2–categorical variant of Lurie’s relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to –categorical localizations, corresponds to Lurie’s scaled unstraightening equivalence. In this –bicategorical context, the relative 2–nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie’s relative nerve when restricted to 1–categories.

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relative nerve, Grothendieck construction, bicategories
Mathematical Subject Classification 2010
Primary: 18D30, 18E35, 18G30, 18G55
Received: 18 October 2019
Revised: 13 January 2020
Accepted: 22 January 2020
Published: 8 December 2020
Fernando Abellán García
Fachbereich Mathematik
Universität Hamburg
Tobias Dyckerhoff
Fachbereich Mathematik
Universität Hamburg
Walker H Stern
Department of Mathematics
University of Virginia
Charlottesville, VA
United States