Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C.
Thm (Rezk): A presentable model category E is a (Grothendieck) homotopy topos iff it has descent.
Note: A (Grothendieck) topos is a presentable elementary topos.
Is there some homotopical definition of an elementary topos?
Are there some references for (approaches, attacks...) Elementary "homotopy" topos?
Motivation: Generalize Elementary "homotopy" topos to Elementary higher topos in analogy to Higher topos theory (à la Lurie, Rezk).
Sheaves in geometry and logic, an intro to topos theory. Maclane and Moerdijk.
Toposes and homotopy toposes, Rezk.