Filter Quotients and Non-Presentable $(\infty ,1)$-Toposes

Abstract

We define filter quotients of $(\infty ,1)$-categories and prove that filter quotients preserve the structure of an elementary $(\infty ,1)$-topos and in particular lift the filter quotient of the underlying elementary topos. We then specialize to the case of filter products of $(\infty ,1)$-categories and prove a characterization theorem for equivalences in a filter product. Then we use filter products to construct a large class of elementary $(\infty ,1)$-toposes that are not Grothendieck $(\infty ,1)$-toposes. Moreover, we give one detailed example for the interested reader who would like to see how we can construct such an $(\infty ,1)$-category, but would prefer to avoid the technicalities regarding filters.

33 Pages, Final version