#### 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
##### Abstract

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 $\infty$–categorical localizations, corresponds to Lurie’s scaled unstraightening equivalence. In this $\infty$–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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