On the $\infty$-categorical Whitehead theorem and the embedding of quasicategories in prederivators

TL;DR

This paper demonstrates that small quasicategories can be embedded into prederivators, which serve as a model for (∞,1)-categories, and proves a Whitehead theorem showing prederivators detect equivalences between large quasicategories.

Contribution

It establishes an embedding of small quasicategories into prederivators and proves a Whitehead theorem for detecting equivalences between large quasicategories.

Findings

Small quasicategories embed into prederivators both simplicially and 2-categorically.

Prederivators can serve as a model for (∞,1)-categories.

Prederivators detect equivalences between arbitrarily large quasicategories.

Abstract

We show that small quasicategories embed, both simplicially and 2-categorically, into prederivators defined on arbitrary small categories, so that in some senses prederivators can serve as a model for $(\infty,1)$-categories. The result for quasicategories that are not necessarily small, or analogously for small quasicategories when mapped to prederivators defined only on finite categories, is not as strong. We prove, instead, a Whitehead theorem that prederivators (defined on any domain) detect equivalences between arbitrarily large quasicategories.