Appears in collection : Toposes Online

Joint work with Mathieu Anel, Eric Finster, and André Joyal

Even though on the surface the theories look similar, there are basic differences between the classical theory of 1-topoi and the theory of $\infty$-topoi. Perhaps the most important difference is that Grothendieck topologies and their associated sheafification functors do not suffice to describe all left exact localizations of a higher presheaf topos. So what is a sheaf in higher topos theory? We answer this question.

We show how to generate the left exact localization of an $\infty$-topos along an arbitrary set of maps S. The associated local objects are called S-sheaves. We also describe the class of maps inverted by this localization. In the case of a higher presheaf topos we obtain a definition of higher site. In that case, if the set S contains only monomorphisms, our definition reduces to the classical notion of Grothendieck topology and Grothendieck site.