nLab six operations

Idea

General

What is known as (Grothendieck’s) six operations is a formalization of structure that

• assigns to every morphism $f$ of suitable spaces a (derived)direct image/(derived)inverse image adjunction $(f^\ast \dashv f_*)$;

• assigns to every separated morphism a direct image with compact support/Verdier dual adjunction $(f_! \dashv f^!)$.

These are four operations and together with

• a closed symmetric monoidal tensor product/internal hom adjunction $((-)\otimes X \dashv [X,-])$

form six operations.

(All this is usually interpreted as derived functors/(infinity,1)-functors so that for instance in the usual application to derived categories of abelian sheaves the last two operations are really Tor and Ext.)

With a list of compatibility conditions between these (for instance (Cisinski-Déglise 09, p. x) this is a structure of Grothendieck’s six operations.

These consistency conditions include the following:

1. The adjunctions are functorial, hence form 2-functors $f \mapsto f_*$, $f \mapsto f_!$:

2. There is a natural transformation $f_! \to f_*$ which is a natural equivalence when $f$ is a proper map.

3. Beck-Chevalley condition (base change formulas): given a (homotopy) pullback diagram

   $$\array{ && Q_1 \underset{X}{\times} Q_2 \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ Q_1 && && Q_2 \\ & {}_{\mathllap{f}}\searrow && \swarrow_{\mathrlap{g}} \\ && X }$$

   such that $f$ is a separated morphism, then there are natural equivalences

   $$g^\ast \circ f_! \simeq (p_2)_! \circ (p_1)^\ast$$
   $$f^! \circ g_\ast \simeq (p_1)_\ast\circ (p_2)^! \,.$$

4. …

(Cisinski-Déglise 09, p. x)

Morover one imposes a formalization of Verdier duality with dualizing object…

(Cisinski-Déglise 09, p. xi)

Specializations

Often specializations of the general concept play a role:

• Wirthmüller context: $f^! \simeq f^\ast$ and $f^\ast$ is a strong closed monoidal functor (hence the projection formula for $f_!$ holds);

  • transfer context: in addition the projection formula for $f_\ast$ holds;

• Verdier-Grothendieck context: the “projection formula” $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$ holds naturally in $X,Y$:

• Grothendieck context: $f_! \simeq f_\ast$ and the projection formula holds $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$.

Mann’s approach

Mann (Mann 22) has developed an abstract framework for a six-functor formalism that combines the approaches of Liu-Zheng and Gaitsgory-Rozenblyum and applies it to rigid analytic geometry, in particular in combination with condensed mathematics and almost mathematics to construct an $\infty$-category? of derived solid $\mathcal{O}_{X}^{+}/\pi$ almost modules (see also Scholze22).

Definition

Let $C$ be an $\infty$-category admitting finite limits and let $E$ be a class of morphisms stable under pullback and composition, and including all isomorphisms. The symmetric monoidal $\infty$-category of correspondences $\mathrm{Corr}(C,E)$ has as its objects the objects in $C$, and has as its morphisms the correspondences in $C$, i.e. for objects $X$ and $Y$ of $C$ a morphism from $X$ to $Y$ is given by an object $W$ of $C$ together with maps

$$\array{ && W \\ & \swarrow_{f} && \searrow_{g} \\ X &&&& Y }$$

where $g\in E$. The symmetric monoidal structure comes from the one on $C$.

Definition

Let $\mathrm{Cat}_{\infty}$ be the $\infty$-category of $\infty$-categories. A 3-functor formalism is a lax symmetric monoidal functor

$$\mathcal{D}:\mathrm{Corr}(C,E)\to \mathrm{Cat}_{\infty}$$

The above definition formalizes three of the functors in the 6-functor formalism. The derived tensor product comes from the lax symmetric monoidal structure, $f^{*}$ comes from the image of the correspondence $Y\xleftarrow{f}X=X$, and $f_{!}$ comes from the image of the correspondence $X=X\xrightarrow{f}Y$.

Definition

A 6-functor formalism is a 3-functor formalism for which $-\otimes A$, $f^{*}$, and $f_{!}$ admit right adjoints.

Properties

Relation to motivic homotopy theory

The initial object in the (infinity,2)-category of functors to stable (infinity,1)-categories which satisfy the six operations formalism (and a bit more, such that A1-homotopy invariance) is stable motivic homotopy theory. See there for more.

Related concepts

• Grothendieck duality, Verdier duality

twisted generalized cohomology theory is conjecturally ∞-categorical semantics of linear homotopy type theory:

linear homotopy type theory	generalized cohomology theory	quantum theory
linear type	(module-)spectrum
multiplicative conjunction	smash product of spectra	composite system
dependent linear type	module spectrum bundle
Frobenius reciprocity	six operation yoga in Wirthmüller context
dual type (linear negation)	Spanier-Whitehead duality
invertible type	twist	prequantum line bundle
dependent sum	generalized homology spectrum	space of quantum states (“bra”)
dual of dependent sum	generalized cohomology spectrum	space of quantum states (“ket”)
linear implication	bivariant cohomology	quantum operators
exponential modality		Fock space
dependent sum over finite homotopy type (of twist)	suspension spectrum (Thom spectrum)
dualizable dependent sum over finite homotopy type	Atiyah duality between Thom spectrum and suspension spectrum
(twisted) self-dual type	Poincaré duality	inner product
dependent sum coinciding with dependent product	ambidexterity, semiadditivity
dependent sum coinciding with dependent product up to invertible type	Wirthmüller isomorphism
$(\sum_f \dashv f^\ast)$-counit	pushforward in generalized homology
(twisted-)self-duality-induced dagger of this counit	(twisted-)Umkehr map/fiber integration
linear polynomial functor	correspondence	space of trajectories
linear polynomial functor with linear implication	integral kernel (pure motive)	prequantized Lagrangian correspondence/action functional
composite of this linear implication with daggered-counit followed by unit	integral transform	motivic/cohomological path integral
trace	Euler characteristic	partition function

References

General abstract discussion is in:

• Halvard Fausk, Po Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories, 11 4 (2003) 107-131 [tac:11-04, pdf]

• Roy Joshua, Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories (pdf)

• Paul Balmer, Ivo Dell'Ambrogio, Beren Sanders, Grothendieck-Neeman duality and the Wirthmüller isomorphism, Compositio Math. 152 (2016) 1740-1776 [arXiv:1501.01999, doi:10.1112/S0010437X16007375]

Lecture notes:

• Martin Gallauer, An introduction to six-functor formalism, lecture at The Six-Functor Formalism and Motivic Homotopy Theory, Università degli Studi di Milano (Sept. 2021) [arXiv:2112.10456, pdf]

• Peter Scholze, Six Functor Formalisms, lecture notes (2022) [pdf, pdf]

• Achim Krause, Thomas Nikolaus, Sheaves on manifolds (2024) [pdf, pdf]

An approach to six functor formalism as a bifibered multicategory (multiderivator, when appropriate) over correspondences is in

• Fritz Hörmann, Fibered derivators, (co)homological descent, and Grothendieck’s six functors pdf;

  Fibered Multiderivators and (co)homological descent, arxiv./1505.00974;

  Six Functor Formalisms and Fibered Multiderivators arXiv:1603.02146;

  Derivator Six Functor Formalisms — Definition and Construction I arXiv:1701.02152

The traditional applications (quasicoherent sheaves of modules) are discussed in

• Joseph Lipman, Notes on derived functors and Grothendieck duality, in Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Math. 1960 Springer (2009) 1-259 [pdf]

• Yves Laszlo, Martin Olsson, The six operations for sheaves on Artin stacks I: Finite Coefficients (arXiv:math/0512097)

An enhanced version of the six operations formalism for etale cohomology of (higher) Artin stacks, using the language of stable (infinity,1)-categories:

• Yifeng Liu, Weizhe Zheng, Enhanced six operations and base change theorem for Artin stacks, arXiv.

The six functor formalism for motivic homotopy theory:

• Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescants dans le monde motivique PhD thesis, Paris [pdf, K:0761]

• Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I), Astérisque 314 (2007) [numdam:AST_2007__314__R1_0]

• Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Astérisque 315 (2007) [numdam:AST_2007__315__1_0]

• Denis-Charles Cisinski, Frédéric Déglise, section A.5 of: Triangulated categories of mixed motives, Springer Monographs in Mathematics, Springer (2019) [arXiv:0912.2110, doi:10.1007/978-3-030-33242-6]

and in the context of rigid analytic geometry:

• Joseph Ayoub, Martin Gallauer, Alberto Vezzani, The six-functor formalism for rigid analytic motives, Forum of Mathematics, Sigma 10 (2022) E61 [doi:10.1017/fms.2022.55, arXiv:2010.15004]

Enhancement to equivariant motivic homotopy theory:

• Marc Hoyois, The six operations in equivariant motivic homotopy theory, Adv. Math. 305 (2017) 197-279 [arXiv:1509.02145, doi:10.1016/j.aim.2016.09.031]

Discussion of six-functor formalism for abelian sheaves on manifolds/topological spaces:

• Masaki Kashiwara, Pierre Schapira, Chapters II and III of: Sheaves on manifolds, Grundlehren 292, Springer (1990) [doi:10.1007/978-3-662-02661-8]

• Marco Volpe, The six operations in topology [arXiv:2110.10212]

See also:

• Brad Drew, Martin Gallauer, The universal six-functor formalism [arXiv:2009.13610]

Discussion for pull-push of (holonomic) D-modules is in

• Joseph Bernstein, around p. 18 of Algebraic theory of D-modules (pdf, ps, dvi)

reviewed for instance in

• Pavel Etingof, Formalism of six functors on all (coherent) D-modules (pdf)

• David Ben-Zvi, David Nadler, section 3 of The Character Theory of a Complex Group (arXiv:0904.1247)

The six operations for derived categories of quasi-coherent sheaves, ind-coherent sheaves, and D-modules on derived stacks are developed in

• Dennis Gaitsgory, Notes on geometric Langlands, web.

Six operations in the setup of o-minimal structures is discussed in

• Mario J. Edmundo, Luca Prelli, The six Grothendieck operations on o-minimal sheaves, Comptes Rendus Mathematique 352:6 (2014) 455-458 arxiv/1401.0846 doi

Grothendieck six operations formalism using diagrams of topoi

• Alexey Kalugin, On categorical approach to Verdier duality, arxiv/1505.06922

Six operations formalism for rigid analytic geometry is in

• Lucas Mann, A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry (arXiv:2206.02022)

The abstract approach behind Mann’s work is the subject of Scholze (2022) and developed in

• Claudius Heyer, Lucas Mann, 6-Functor Formalisms and Smooth Representations [arXiv:2410.13038]