**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.