**Abstract: **The Yoneda philosophy tells us that studying a small category is equivalent to studying set-valued functors on it. Joyal, in a letter to Grothendieck, outlined a strategy for studying algebraic varieties by considering presheaves valued not in sets, but rather in simplicial sets. This was built on work of Illusie using the technology of model categories developed earlier by Quillen. This marriage of ideas could be considered the first true appearance of a "homotopy theory of algebraic varieties," known now by the much snappier name "local homotopy theory." In this talk we will present a friendly introduction to local homotopy theory, focusing on the topic of descent along Galois field extensions, and time permitting we will discuss a version of the Lichtenbaum-Quillen conjecture.

### Graduate Student Geometry-Topology Seminar

Monday, September 19, 2022 - 4:30pm

#### Thomas Brazelton

UPenn