Fibered Multiderivators and (co)homological descent

Abstract

The theory of derivators enhances and simplifies the theory of triangulated categories. In this article a notion of fibered (multi-)derivator is developed, which similarly enhances fibrations of (monoidal) triangulated categories. We present a theory of cohomological as well as homological descent in this language. The main motivation is a descent theory for Grothendieck's six operations. NotationWe denote by CAT the 2-"category" 1 of categories, by (S)MCAT the 2-"category" of (symmetric) multicategories, and by Cat the 2-category of small categories. We consider a partially ordered set (poset) X as a small category by considering the relation x ≤ y to be equivalent to the existence of a unique morphism x → y. We denote the positive integers (resp. non-negative integers) by N (resp. N 0 ). The ordered sets {0, . . . , n} ⊂ N 0 considered as a small category are denoted by ∆ n . We denote by Mor(D) (resp. Iso(D)) the class of morphisms (resp. isomorphisms) in a category D. The final category (which consists of only one object and its identity) is denoted by ⋅ or ∆ 0 . The same notation is also used for the final multi-category, i.e. that with one object and precisely one n-ary morphism for any n. Our conventions about multicategories and fibered (multi-)categories are summarized in appendix A.

Six-functor-formalisms and fibered multiderivators

“…In [11] we showed that all isomorphisms of 0.1 follow from this definition. We explain in Section 8 that enlarging the domain of 2-morphisms to all morphisms, or to a class of "proper", or "etale" morphisms, respectively, one can easily encode all sorts of more strict sixfunctor-formalisms, where either f !…”

“…For a detailed introduction to derivators and fibered multiderivators we refer to [11]. Stable derivators, among other things, simplify, enhance and conceptually explain triangulated categories.…”

“…Let D and S be pre-multiderivators satisfying (Der1) and (Der2) (cf. [11,Definition 1.3.5.]). A strict morphism of pre-multiderivators D → S is a left (resp.…”

“…2.16) between 1-opfibrations (and 2-fibrations) of 2-multicategories over Dia cor (S) with 1-categorical fibers and pseudo-functors Dia cor (S) → CAT into the 2-multicategory of all categories (where the multi-structure is given by multivalued functors) this formulation unifies in a nice way the two pseudofunctors Dia(S) 1−op → CAT resp. Dia op (S) 1−op → CAT that have been associated with a fibered multiderivator in [11], because there are embeddings of Dia(S) 1−op and Dia op (S) 1−op into Dia cor (S) (cf. 5.8-5.9).…”

Descent for coherent sheaves along formal/open coverings

“…a small category) and a functor S ∶ I → S form a 2-category Dia op (S), called the category of diagrams in S (cf. also [5]). A morphism of S-diagrams (α, µ)…”