Topology Seminar: Marc Hoyois

The formalism of six operations encodes the functorial behavior of (co)homology theories. It was first introduced by Grothendieck for the l-adic cohomology of schemes, and was later developed in a variety of other geometric contexts: D-modules on schemes, spectra parametrized by topological spaces, motivic spectra parametrized by schemes, etc. Equivariant homotopy theory is also best understood as a formalism of six operations for topological stacks. In this talk I will discuss the basics and the significance of this formalism, and I will then describe an extension of motivic homotopy theory to algebraic stacks.