2023/09/20 by Kuijper, Josefien
#14F99 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.2309.11449
In this paper, we consider some variations on Mann's definition ∞-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 ∞-categories, and have recollements. Using Nagata's compactification theorem, we show that Nagata six-functor formalism 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 characterisation, 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 ∞-categories and adjoint triples. We characterise which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable ∞-categories and adjoint triples, extend to local six-functor formalisms.