Uniqueness of six-functor formalisms [OS Arithmetic Geometry and Representation Theory]

In recent years, many efforts have been made to formalise, in the most efficient way,  Grothendieck’s six operations of sheaves (tensor product and internal hom, inverse and direct image, and exceptional inverse and direct image) and their properties. In addition, one might want to encode natural isomorphisms between the inverse image and exceptional inverse image for a certain class of “étale” morphisms, and between the direct image and the exceptional direct image for a class of “proper morphisms”. These can be encoded as extra data, using (infinity,2)-categories, but this is not necessary. More recently, it has become clear that these natural isomorphisms canonically arise from a property of the six operations themselves.
We show how to formulate this condition in an efficient way, leading to the notion of a “Nagata six-functor formalism”. We show that Nagata six-functor formalisms are uniquely determined by the data of the tensor product and the inverse image functor, giving a positive answer to a conjecture by Scholze. This talk is based on joint work with Adam Dauser.