2025 Theses Doctoral

Geometric Criterion for 6-Functor Formalisms: Betti Realization, Gluing, and the Motivic Homotopy Theory of Complex Analytic Stacks

Magen, Roy

The concrete goals of this dissertation are to define the motivic homotopy theory of complex analytic stacks, establish some of its properties, show it admits the structure of a 6-functor formalism, and produce a stacky version of Betti realization that is compatible Grothendieck's six operations. In order to achieve these goals, we develop a framework for studying ``generalized cohomology theories'' in an abstract geometric context, building on [25], and using material on 6-functor formalisms from [62] and [39].

In particular, we give criteria for establishing the structure of a 6-functor formalism, as well as the compatibility of morphisms with the six operations. We also establish strong tools for proving versions of Morel-Voevodsky's localization theorem, which is fundamental for our study of cohomology theories, 6-functor formalisms, and the motivic homotopy theory of complex analytic stacks.