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Filter Quotient ∞-Categories - Nima Rasekh

Abstract: Classical topos theory broadly breaks down into two branches: Grothendieck topos theory, with applications in geometry, and elementary topos theory, with applications in logic. Grothendieck topos theory has already been generalized to the higher categorical setting, with applications ranging from derived geometry to Goodwillie calculus. On the other hand, the elementary aspects of higher topos theory have been recently on the rise with connections to elementary topos theory and homotopy type theory.

In this talk we introduce the filter quotient construction for ∞-categories with a chosen filter of subobjects. We will then focus on the particular case of filter products. Using filter products we will construct examples of higher toposes that are not Grothendieck, in the sense that they are not presentable and do not even have countable coproducts, and look at its implications on non-standard truncations levels.

If time permits, we will discuss how we can use the filter product construction to construct new models of homotopy type theory in the sense of Shulman and Lumsdaine, using model categorical techniques.