TITLE: What is an Elementary "Homotopy, Model" Topos?
QUESTION [11 upvotes]: 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.

REPLY [9 votes]: Since the time when Denis referred in the comments to the relevant nLab page, there has been a new proposal written up by Mike Shulman there:
An elementary $(\infty,1)$-topos is an $(\infty,1)$-category $\mathbf{E}$ such that

$\mathbf{E}$ has finite (∞,1)-limits and colimits
$\mathbf{E}$ is locally cartesian closed
There exists a subobject classifier (an object classifier that classifies the collection of all monomorphisms in an (∞,1)-category)
For any morphism $f:Y\to X$ in $\mathbf{E}$, there exists an object classifier in $\mathbf{E}$ classifying a class of morphisms that (1) includes $f$ and (2) is closed under fiberwise finite limits and colimits, composition (i.e. dependent sums), and dependent products.

For discussion see at the n-Category Café here.