I've been reading through Lurie's book on higher topos theory, where he develops the theory of $(\infty,1)$-toposes, which leads me to the following question: Is there any sort of higher topos theory on the more general $\omega$-categories, where we don't require all higher morphisms to be invertible?



$\omega$-topos theory?
The short answer is no. Even 2-toposes are poorly understood -- we don't know what the right definition is. For higher dimensions, including $\infty$, we *definitely* don't have the answers.


Just as the primordial example of a (1-)topos is $\mathbf{Set}$ (the 1-category of sets and functions), the primordial example of a 2-topos should be $\mathbf{Cat}$ (the 2-category of categories, functors and natural transformations). 


Mark Weber has done some work on 2-toposes, building on earlier ideas of Ross Street. But I think Mark is quite open about the tentative nature of this so far. 


There was a [good discussion](http://golem.ph.utexas.edu/category/2008/01/2toposes.html) of the current state of 2-toposes (and more generally n-toposes) at the $n$-Category Café last year: 


