Contents

Ingredients

Concepts

Constructions

Examples

Theorems

cohomology

# Contents

## Idea

### General

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

These are four operations and together with

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

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

Morover one imposes a formalization of Verdier duality with dualizing object

### 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);

• 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 (Mann22) 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 $g\inE$. 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.

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

linear homotopy type theorygeneralized cohomology theoryquantum theory
linear type(module-)spectrum
multiplicative conjunctionsmash product of spectracomposite system
dependent linear typemodule spectrum bundle
Frobenius reciprocitysix operation yoga in Wirthmüller context
invertible typetwistprequantum line bundle
dependent sumgeneralized homology spectrumspace of quantum states (“bra”)
dual of dependent sumgeneralized cohomology spectrumspace of quantum states (“ket”)
linear implicationbivariant cohomologyquantum operators
exponential modalityFock space
dependent sum over finite homotopy type (of twist)suspension spectrum (Thom spectrum)
dualizable dependent sum over finite homotopy typeAtiyah duality between Thom spectrum and suspension spectrum
(twisted) self-dual typePoincaré dualityinner product
dependent sum coinciding with dependent productambidexterity, semiadditivity
dependent sum coinciding with dependent product up to invertible typeWirthmüller isomorphism
$(\sum_f \dashv f^\ast)$-counitpushforward in generalized homology
(twisted-)self-duality-induced dagger of this counit(twisted-)Umkehr map/fiber integration
linear polynomial functorcorrespondencespace of trajectories
linear polynomial functor with linear implicationintegral kernel (pure motive)prequantized Lagrangian correspondence/action functional
composite of this linear implication with daggered-counit followed by unitintegral transformmotivic/cohomological path integral
traceEuler characteristicpartition function

General abstract discussion is in

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., no. 1960, Springer-Verlag, New York, 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 Artin stacks (and higher Artin stacks?), using the language of stable (infinity,1)-categories is developed in

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

The six functor formalism for motivic homotopy theory was developed in

• Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescants dans le monde motivique PhD thesis, Paris (pdf)

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

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

reviewed for instance in

A quick list of the axioms with a Grothendieck’s six operations with an eye towards the definition of motives is in section A.5 of

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

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

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 the following course by Peter Scholze:

• Peter Scholze, Six Functor Formalisms pdf