Higher stacks as diagrams

Abstract

Several possible presentations for the homotopy theory of (non-hypercomplete) ∞-stacks on a classical site S are discussed. In particular, it is shown that an elegant combinatorial description in terms of diagrams in S exists, similar to Cisinski's presentation, based on work of Quillen, Thomason and Grothendieck, of usual homotopy theory by small categories and their smallest (basic) localizer. As an application it is shown that any (local) fibered (a.k.a. algebraic) derivator over S with stable fibers extends to ∞-stacks in a well-defined way under mild assumptions.

Six-Functor-Formalisms on Higher Stacks

“…This is dual to Proposition 5.9 except for the multi-aspect. However, the same reasoning as in [26,Proposition 8.10] works.…”

“…As in [26,Proposition 8.11], one can instead construct a fibered derivator D !,h → H 2 (M), where H 2 (M) is the homotopy 2-pre-derivator associated with M. We will not use this because a similar extension can also be extracted from an extended * , !-formalism. (Co)homological descent implies that the restriction of D !,h along S → M is equivalent to D !…”

“…is an equivalence of categories with weak equivalences with quasi-inverse given by the nerve, and by e.g. [26,Proposition A.6] the homotopy colimit for the pair (Dia(S), W ∞ ) is given by the Grothendieck construction. For a morphism α ∶ (I, S) → (J, T ) in Dia(M) with the property that hocolim I S → hocolim J T is a weak equivalence, it implies that…”

“…Let X ∈ M. In SET S op ×∆ op there is a morphism X → X (cofibrant replacement in the projective model structure) with X ∈ S ∐,∆ op . This yields a morphism 26…”

“…The objective of this article is to find necessary conditions under which the six-functor-formalism extends to M loc (or to a suitable subcategory) such that for X ∈ M loc the category D X does -up to equivalence -only depend on the weak equivalence class of X, and to construct isomorphisms between the six functors analogous to (1). It was explained in [26] that, at least if there is a derivator enhancement of the six-functor-formalism with stable and wellgenerated fibers, there are two potential ways of defining D X , one using cohomological descent for the * -functors, and one using homological descent for the !-functors. Informally, those categories are defined as follows: Since S is small, one can represent X by a simplicial object X• ∈ S ∐,∆ op .…”