An axiomatization of six-functor formalisms

Abstract In this paper, we consider two variations on Mann’s $\infty $ -categorical definition of abstract six-functor formalisms. We consider Nagata six-functor formalisms, that have the additional requirement of having Grothendieck and Wirthmüller contexts. We also consider local six-functor formalisms, which in addition to this, take values in presentable stable $\infty $ -categories, and have recollements. Using Nagata’s compactification theorem, we show that Nagata six-functor formalisms on varieties can be given by just specifying adjoint triples for open immersions and for proper morphisms, satisfying certain compatibilities. The existence of recollements is (almost) equivalent to a hypersheaf condition for a Grothendieck topology on the category of “varieties and spans consisting of an open immersion and a proper map”. Using this characterization, we show that the category of local six-functor formalisms embeds faithfully into the category of lax symmetric monoidal functors from the category of smooth and complete varieties to the category of presentable stable $\infty $ -categories and adjoint triples. We characterize which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable $\infty $ -categories and adjoint triples, extend to local six-functor formalisms.