higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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: 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 $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$.
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 ∞-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)
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
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
Six operations in the setup of o-minimal structures is discussed in