A higher Whitehead theorem and the embedding of quasicategories in prederivators

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.

Keywords

prederivator, quasicategory, models for higher categories

2010 Mathematics Subject Classification

18G55, 55U35

Full Text (PDF format)

Received 24 August 2018

Received revised 27 May 2019

Accepted 9 March 2019

Published 6 November 2019