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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 $g\inE$. 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
H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (TAC, 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, arXiv:1501.01999.
An approach to six functor formalism as a bifibered multicategory (multiderivator, when appropriate) over correspondences is in
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
The six functor formalism for motivic homotopy theory was developed in
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)
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
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 the following course by Peter Scholze:
Last revised on February 28, 2023 at 08:31:40. See the history of this page for a list of all contributions to it.