A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi

TL;DR

This paper characterizes simplicial categories arising from closed model categories as $ abla$-pretopoi, extending Giraud's theorem to the $"infty$-categorical

Contribution

It provides a Giraud-type characterization of $"infty$-pretopoi, linking model categories, homotopy colimits, and Segal categories.

Findings

Equivalence between model categories and $"infty$-pretopoi

Identification of conditions for a simplicial category to be an $"infty$-pretopos

Extension of Giraud's theorem to the $"infty$-categorical

Abstract

Theorem (after Giraud, SGA 4): Suppose $A$ is a simplicial category. The following conditions are equivalent: (i) There is a cofibrantly generated closed model category $M$ such that $A$ is equivalent to the Dwyer-Kan simplicial localization $L(M)$; (ii) $A$ admits all small homotopy colimits, and there is a small subset of objects of $A$ which are $A$-small, and which generate $A$ by homotopy colimits; (iii) There exists a small 1-category $C$ and a morphism $g:C\to A$ sending objects of $C$ to $A$-small objects, which induces a fully faithful inclusion $i:A\to \hat{C}$, such that $i$ admits a left homotopy-adjoint $\psi$. We call a Segal category $A$ which satisfies these equivalent conditions, an $\infty$-pretopos. Note that (i) implies that $A$ admits all small homotopy limits too. If furthermore there exists $C\to A$ as in (iii) such that the adjoint $\psi$ preserves finite homotopy limits, then we say that $A$ is an ``$\infty$-topos''.