Title: A Giraud-type characterization of the simplicial categories
associated to closed model categories as $\infty$-pretopoi
Author: Carlos Simpson
Address: CNRS UMR 5580, Laboratoire Emile Picard,
Universite Paul Sabatier, 31062 Toulouse CEDEX, France
Email: carlos@picard.ups-tlse.fr
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 $g:C\rightarrow A$
sending objects of $C$ to $A$-small objects, which induces a fully faithful
inclusion $i:A\rightarrow \widehat{C}$, such that $i$ admits a left
homotopy-adjoint $\psi$.
We call a Segal category $A$ which satisfies these equivalent conditions,
an {\em $\infty$-pretopos}. Note that (i) implies that $A$ admits all small
homotopy limits too.
If furthermore there exists $C\rightarrow A$ as in (iii) such that the
adjoint $\psi$ preserves finite homotopy limits, then we say that $A$ is an
``$\infty$-topos''.