I will discuss a new infinitycategorical definition of abstract sixfunctor 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 sixfunctor 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 sixfunctor formalisms and compactly supported cohomology. We can show that sixfunctor 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 infinitycategories and adjoint triples, extend to sixfunctor formalisms.
Algebra Seminar
Friday, October 20, 2023  3:30pm
Josefien Kuijper
Stockholm University
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