Geometric Criteria for 6-Functor Formalisms in the Setting of Pullback Formalisms

TL;DR

This work develops a generalized framework for Grothendieck's six operations beyond classical algebraic geometry by introducing pullback formalisms and a robust notion of cohomological properness, gluing, and duality. It proves the stable motivic homotopy theory of algebraic stacks is universal among six-functor formalisms and extends universality results to complex analytic stacks, including a stacky Betti realization compatible with all six operations. Central to the approach is a generalized Voevodsky criterion that applies to broad geometric contexts and guarantees that morphisms of coefficient systems respect six-functor structures. The results yield a powerful, transferable toolkit for constructing and comparing six-functor formalisms across algebraic, analytic, and stacky settings, with significant implications for Betti realizations, ambidexterity, and duality theories. Overall, the paper advances a unifying, axiomatic treatment of six-functor formalisms and their morphisms in diverse geometric contexts, enabling universal properties and concrete realizations in new domains.

Abstract

In this article, we prove various results about six-functor formalisms and morphisms between them. In particular, we show that the stable motivic homotopy theory of algebraic stacks is the universal six-functor functor formalism in a strong sense: it is initial in some category whose objects are six-functor formalisms, and whose morphisms commute with the six operations. This strengthens previous results of this form, which only showed that stable motivic homotopy theory is a six-functor formalism, and is initial, but not that the morphisms from it commute with the six operations. We also prove a similar result for the stable motivic homotopy theory of complex analytic stacks, and produce a Betti realization for stacks that commutes with Grothendieck's six operations, generalizing previous results of Ayoub for quasi-projective schemes. In order to do this, we give a generalized and enhanced account of Voevodsky's geometric criterion for six-functor formalisms. Our version of Voevodsky's principle makes sense in more general geometric contexts (not only algebraic geometry), and also provides a criterion for showing that a morphism between six-functor formalisms is compatible with the six operations.

Theorems & Definitions (181)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.4
• Definition 1.5
• Theorem 1.6
• Proposition 1.7
• Proposition 1.8
• Theorem 1.9
• Theorem 1.10