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

In this article, we study criteria for producing six-functor formalisms and morphisms between them. One notable application is that the 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 all six operations. As a further application, we produce an analytic realization to a complex analytic version of motivic homotopy theory that is compatible with the six operations, and extend Betti realization to a map from this complex analytic version that is also compatible with the six operations. The abstract nature of our results is suitable for applications to many geometric contexts, allowing us to prove a similar result for the motivic homotopy theory of complex analytic stacks as a six-functor formalism defined on complex analytic stacks. Our main general result is a generalized and enhanced version of Voevodsky's geometric criterion for six-functor formalisms, given in terms of localization and duality properties. Our version of Voevodsky's principle makes sense in very general geometric contexts, and provides criteria not only for showing when presheaves extend to six-functor formalism, and when a transformation between six-functor formalisms is compatible with the six operations, but also for when a transformation to an ordinary presheaf extends to a morphism of six-functor formalisms (and therefore establishing the six operations for the codomain).