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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
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:
The adjunctions are functorial, hence form 2-functors $f \mapsto f_*$, $f \mapsto f_!$:
There is a natural transformation $f_! \to f_*$ which is a natural equivalence when $f$ is a proper map.
Beck-Chevalley condition (base change formulas): given a (homotopy) pullback diagram
such that $f$ is a separated morphism, then there are natural equivalences
…
Morover one imposes a formalization of Verdier duality with dualizing object…
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 (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).
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
where $g\in E$. The symmetric monoidal structure comes from the one on $C$.
Let $\mathrm{Cat}_{\infty}$ be the $\infty$-category of $\infty$-categories. A 3-functor formalism is a lax symmetric monoidal functor
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 $X=X\xrightarrow{f}Y$.
A 6-functor formalism is a 3-functor formalism for which $-\otimes A$, $f^{*}$, and $f_{!}$ admit right adjoints.
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:
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]
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:
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:
Enhancement to equivariant motivic homotopy theory:
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:
Discussion for pull-push of (holonomic) D-modules is in
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
Six operations in the setup of o-minimal structures is discussed in
Grothendieck six operations formalism using diagrams of topoi
Six operations formalism for rigid analytic geometry is in
The abstract approach behind Mann’s work is the subject of Scholze (2022).
Last revised on May 19, 2023 at 16:50:25. See the history of this page for a list of all contributions to it.