Toposes Online

Collection Toposes Online

Organizer(s) Olivia Caramello, Alain Connes, Laurent Lafforgue
Date(s) 28/06/2021 - 30/06/2021
linked URL https://aroundtoposes.com/toposesonline/
00:00:00 / 00:00:00
17 31

Higher Sheaves

By Georg Biedermann

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.

Information about the video

  • Date of recording 28/06/2021
  • Date of publication 28/06/2021
  • Institution IHES
  • Language English
  • Audience Researchers
  • Format MP4

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