Higher geometric sheaf theories

The notion of a geometric category is a categorification of the notion of a topological space in as much as it it is exactly the notion of a categorical model of a first order geometric theory. Equipped with its canonical structure as a site, it is an essential notion in topos theory as it is a pivotal point between sheaf theory and symbolic logic.

In this talk, we introduce an infinity-category of geometric infinity-categories which appears to play a role in infinity-topos theory parallel to the one that the (2-)category of ordinary geometric categories plays in ordinary topos theory. In particular, we introduce a canonical higher geometric sheaf theory on such geometric infinity-categories which faithfully generalizes Lurie's sheaf theories over infinity-toposes, but which generally differs from the topological sheaf theory generated by ordinary geometric covers. Eventually, we discuss some applications and expectations.