Context:
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.
Questions
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).
References:
Sheaves in geometry and logic, an intro to topos theory. Maclane and Moerdijk.
Toposes and homotopy toposes, Rezk.