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

Kevin Arlin

TL;DR

The paper investigates how small and large quasicategories (i.e., $( rightarrow)$-categories) embed into prederivators, providing a bridge between $ extbf{QCAT}$ and derivator-based models. It constructs the homotopy prederivator $ ext{HO}(Q)$ from a quasicategory $Q$ and proves both a simplicial and a 2-categorical fully faithful embedding of quasicategories into prederivators, with a bicategorical conservativity result that prederivators detect equivalences of quasicategories of any size. A key technical tool is Joyal–Stevenson delocalization, showing every quasicategory is a localization of a category, which underpins the 2-categorical embedding and the Whitehead-type theorem. The results collectively demonstrate that prederivators are a robust model for $( rightarrow)$-categories and can faithfully encode their equivalences and mapping spaces, advancing the understanding of how $ extbf{QCAT}$ sits inside derivator frameworks.

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.

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

TL;DR

The paper investigates how small and large quasicategories (i.e., -categories) embed into prederivators, providing a bridge between and derivator-based models. It constructs the homotopy prederivator from a quasicategory and proves both a simplicial and a 2-categorical fully faithful embedding of quasicategories into prederivators, with a bicategorical conservativity result that prederivators detect equivalences of quasicategories of any size. A key technical tool is Joyal–Stevenson delocalization, showing every quasicategory is a localization of a category, which underpins the 2-categorical embedding and the Whitehead-type theorem. The results collectively demonstrate that prederivators are a robust model for -categories and can faithfully encode their equivalences and mapping spaces, advancing the understanding of how sits inside derivator frameworks.

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 -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.

Paper Structure

This paper contains 6 sections, 16 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Background on 2-categories and prederivators
  3. The basic construction
  4. The simplicial embedding $\mathbf{QCAT}_\bullet\to\mathbf{PDer}_\bullet$
  5. The embedding $\underline{\mathbf{QCat}}\to\underline{\mathbf{PDer}}$ of 2-categories
  6. Whitehead's theorem for quasicategories

Key Result

Lemma 8

The equivalence class $[\alpha]$ of a map $\alpha:Q\to R^{[1]}$ is an isomorphism in the homotopy category $\mathrm{Ho}(R^Q)$ if and only if, for every vertex $q\in Q_0$ of $Q$, the equivalence class $[\alpha(q)]$ is an isomorphism in $\mathrm{Ho}(R)$.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Lemma 8: riehl, 2.3.10
  • Definition 9
  • Lemma 10
  • ...and 36 more