Six Functor Formalisms and Fibered Multiderivators

Hörmann, Fritz

doi:10.48550/arxiv.1603.02146

Cited by 3 publications

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“…We assume that S is already defined on the larger diagram category Dia and that D is a left (resp. right) fibered multiderivator over S such that also (FDer0 right) 4 resp. (FDer0 left) hold true.…”

Section: Introductionmentioning

confidence: 99%

Enlargement of (fibered) derivators

Hörmann

2020

Journal of Pure and Applied Algebra

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We show that the theory of derivators (or, more generally, of fibered multiderivators) on all small categories is equivalent to this theory on partially ordered sets, in the following sense: Every derivator (more generally, every fibered multiderivator) defined on partially ordered sets has an enlargement to all small categories that is unique up to equivalence of derivators. Furthermore, extending a theorem of Cisinski, we show that every bifibration of multi-model categories (basically a collection of model categories, and Quillen adjunctions in several variables between them) gives rise to a left and right fibered multiderivator on all small categories.1 Small categories Along the lines we also prove that a left (or right) fibered multiderivator defined on a smaller diagram category can be intrinsically enlarged to a larger diagram category, whenever some nervelike construction is available, relating the two diagram categories. This is more of theoretical interest because most derivators occurring in nature come from model categories. One concrete result of the construction is the following: Corollary 1.2. Let D → S be a left (resp. right) fibered multiderivator with domain Invpos 2 (resp. Dirpos 3 ) such that S is defined on Cat and such that also (FDer0 right) (resp. (FDer0 left)) holds. Then there exists an enlargement of D to a left (resp. right) fibered multiderivator E → S with domain Cat, such that its restriction to Invpos (resp. Dirpos) is equivalent to D. Any other such enlargement is equivalent to E.Note that this holds, in particular, for usual derivators (take for S the final pre-derivator) and closed monoidal derivators (take for S the final pre-multiderivator).Proof. In the left case, apply the machine of Theorem 4.1 twice using the functors N constructed in Proposition 2.5, firstly for the pair (Invpos ⊂ Cat ○ ), and secondly for the pair (Inv ⊂ Cat). Similarly for the right case.If we start with a fibered multiderivator on all of Pos, however, we show that the two extensions to Cat agree. Therefore we arrive at the following Corollary 1.3. Let D → S be a left and right fibered multiderivator with domain Pos such that S is defined on Cat. Then there exists a canonical enlargement of D to a left and right fibered multiderivator E → S with domain Cat, such that its restriction to Pos is equivalent to D. The enlargement is unique up to equivalence of fibered multiderivators over S.Actually, here Pos can be even replaced by the smallest diagram category containing both Invpos and Dirpos.Proof. We consider this time the pairs (Pos ⊂ Cat ○ ) and (Cat ○ ⊂ Cat). For each of these pairs we dispose of functors N as in 2.3 for the left and the right case simultaneously by Proposition 2.5 (by enlarging Dia ′ we only weaken the axioms). Hence we may conclude by applying Proposition 4.10 twice.For the reader mainly interested in plain (left and right) derivators, we state explicitly: Corollary 1.4. Let D be a derivator with domain Pos. Then there exists a canonical enlargement of D to a derivator E with doma...

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Section: Introductionmentioning

confidence: 99%

Enlargement of (fibered) derivators

Hörmann

2020

Journal of Pure and Applied Algebra

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“…Of course, there have to be compatibilities among those natural isomorphisms, and it is not easy to give a complete list of them. In [4] we explained how to proceed in a more abstract way (like in the ideas of fibered category or multicategory) and get a precise definition of a six-functor-formalism without having to specify any of these compatibilities explicitly.…”

Section: Introductionmentioning

confidence: 99%

“…We now recall the precise definition of six-functor-formalisms from [4]. Let S be a (base) category with fiber products as above.…”

Section: Introductionmentioning

confidence: 99%

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Derivator Six Functor Formalisms -- Definition and Construction I

Hörmann

2017

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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.

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“…On the higher categorical level, six functor formalisms have been considered in the context of stable ∞-categories by Liu and Zheng [LZ17] and in the context of derivators by Hörmann [Hör17]. The notion of a derived functor in the dg setting is discussed by Drinfeld [Dri08].…”

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Six operations on dg enhancements of derived categories of sheaves

Schnürer¹

2015

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We lift Grothendieck-Verdier-Spaltenstein's six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. Our main tools come from enriched model category theory. Contents 1. Introduction 1 2. Description of the main results 9 3. Dg k-enriched functorial factorizations 17 4. Applications 24 5. Some 2-multicategories 37 6. Four operations 46 7. Two operations 67 8. The 2-multicategory of formulas 84 9. Some remarks on 2-morphisms in ENH k 87 Appendix A. Spaltenstein's results generalized to ringed topoi 93 Appendix B. Change of universe 97 Appendix C. Some results on model categories 100 References 104 a cartesian diagram of top. spaces with α separated, loc. proper, and α ! : Mod(Y A ) → Mod(X A ) of finite cohom. dimension.

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Six Functor Formalisms and Fibered Multiderivators

Cited by 3 publications

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Enlargement of (fibered) derivators

Enlargement of (fibered) derivators

Derivator Six Functor Formalisms -- Definition and Construction I

Six operations on dg enhancements of derived categories of sheaves

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