Homology, Homotopy and Applications

Volume 22 (2020)

Number 1

A higher Whitehead theorem and the embedding of quasicategories in prederivators

Pages: 117 – 139

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a8


Kevin Arlin (Department of Mathematics, University of California at Los Angeles)


We prove a categorified Whitehead theorem showing that the $2$-functor $\mathrm{HO}$ associating a prederivator to a quasicategory reflects equivalences. The question of whether $\mathrm{HO}$is bicategorically fully faithful (that is, whether morphisms and $2$-morphisms can be uniquely lifted from prederivators to quasicategories) is more subtle.We can show that small quasicategories embed fully faithfully, both bicategorically and with respect to a certain simplicial enrichment, into prederivators defined on arbitrary small categories. When the quasicategories are not necessarily small, or when the prederivators are defined only on homotopically finite categories, the $2$-categorical argument breaks down, although the simplicial version continues to go through. We give a conjectural counterexample to bicategorical full faithfulness in general.


prederivator, quasicategory, models for higher categories

2010 Mathematics Subject Classification

18G55, 55U35

Copyright © 2019, Kevin Arlin. Permission to copy for private use granted.

Received 24 August 2018

Received revised 27 May 2019

Accepted 9 March 2019

Published 6 November 2019