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

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 $ψ$. 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 $ψ$ preserves finite homotopy limits, then we say that $A$ is an ``$\infty$-topos''.