Derivator Six Functor Formalisms -- Definition and Construction I

Abstract

A theory of a derivator version of six-functor-formalisms is developed, using an extension of the notion of fibered multiderivator due to the author. Using the language of (op)fibrations of 2multicategories this has (like a usual fibered multiderivator) a very neat definition. This definition not only encodes all compatibilities among the six functors but also their interplay with homotopy Kan extensions. One could say: a nine-functor-formalism. This is essential, for instance, to deal with (co)descent questions. Finally, it is shown that every fibered multiderivator (for example encoding any kind of derived four-functor formalism (f * , f * , ⊗, HOM) occurring in nature) satisfying base-change and projection formula formally gives rise to such a derivator six-functor-formalism in which "f ! = f * ", i.e. a derivator Grothendieck context.. This is the classical formula of Serre duality.Example 1.3 (Poincaré duality). Let k be a field. If S is a category of nice topological spaces, we have a six-functor-formalism in which D S is the derived category of sheaves of k-vector spaces on S. Let X be an n-dimensional topological manifold. Consider a local system E of k-vector spaces on X and consider the isomorphism (again one of the two adjoints of the projection formula):, where L or is the orientation sheaf of X over k. Taking i-th homology of complexes we arrive at. This is the classical formula of Poincaré duality. Example 1.4 (Group (co)homology). The six-functor-formalism of Example 1.3 extends to stacks. Let G be a group and consider the classifying stack [⋅ G] and the projection π ∶ [⋅ G] → ⋅. Note: Abelian sheaves on [⋅ G] = G-representations in Abelian groups. The extension of the six-functorformalism encodes duality theorems like Tate duality. In this case π * yields group cohomology and π ! yields group homology. If G is finite, we also have a natural morphism π ! → π * whose cone (homotopy cokernel) is Tate cohomology.

Higher stacks as diagrams

“…Let I be a small category and let M be one of the model categories in 5.1. Then the functor 10 M I S I → S I equipped point-wise with the model category structure of over-category, is a bifibration of model categories. Furthermore the push-forward functors f • and pull-back functors f • preserve weak equivalence (i.e.…”

Derivator Six-Functor-Formalisms -- Construction II

“…Of course this definition depends on the chosen class of proper morphisms. Note that the analog of [13,Lemma 7.10] holds true for weakly type i admissible. Let S now be an opmulticategory (in this article always a usual category S equipped with the opmulticategory structure encoding the product).…”

“…In this section we recall from [13] the notion of 2-pre-multiderivator and fibered multiderivator (with 2-categorical bases).…”

“…In [13] we defined a fibered multiderivator as below. Above we gave the equivalent patchwork definition because the axioms are anyway the ones to be checked.…”

“…For a detailed introduction to classical six-functor-formalisms we refer to the previous article [13].…”