## Algebra Seminar

I will discuss a new infinity-categorical definition of abstract six-functor formalisms. Our definition is a variation on Mann's definition, with the additional requirement of having Grothendieck and Wirthmüller contexts, and recollements. Using Nagata's compactification theorem, we show that such a six-functor formalism can be given by just specifying adjoint triples on open immersions and on proper maps, satisfying certain compatibilities. Moreover, the existence of recollements is equivalent to a sheaf condition for a Grothendieck topology on the category of “varieties and spans with an open immersion and a proper map”. This brings to light an interesting analogy between abstract six-functor formalisms and compactly supported cohomology. We can show that six-functor formalisms, according to our definition, are uniquely determined by the restriction of the inverse image (upper star) to smooth and complete varieties. Moreover we can characterize which lax symmetric monoidal functors from the category of complete varieties to the category of stable infinity-categories and adjoint triples, extend to six-functor formalisms.