group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Fourier-Mukai transform is a categorified integral transform analogous to the standard Fourier transform.
Let $X$ and $Y$ be schemes over a field $K$. Let $E \in D(QCoh(O_{X \times Y}))$ be an object in the derived category of quasi-coherent sheaves over their product. (This is a correspondence between $X$ and $Y$ equipped with a chain complex $E$ of quasi-coherent sheaves).
The functor $\Phi(E) : D(QCoh(O_X)) \to D(QCoh(O_Y))$ defined by
where $p$ and $q$ are the projections from $X \times Y$ onto $X$ and $Y$, respectively, is called the Fourier-Mukai transform of $E$, or the Fourier-Mukai functor induced by $E$.
When $F : D(QCoh(O_X)) \to D(QCoh(O_Y))$ is isomorphic to $\Phi(E)$ for some $E \in D(QCoh(O_{X \times Y}))$, one also says that $F$ is represented by $E$ or simply that $F$ is of Fourier-Mukai type.
The key fact is as follows
Let $X$ and $Y$ be smooth projective varieties over a field $K$. Let $F : D(X) \to D(Y)$ be a triangulated fully faithful functor. Then $F$ is represented by some object $E \in D(X \times Y)$ which is unique up to isomorphism.
See Orlov 2003, 3.2.1 for a proof. Though the theorem is stated there for $F$ admitting a right adjoint, it follows from Bondal-van den Bergh 2002 that every triangulated fully faithful functor admits a right adjoint automatically.
It is generally believed that this theorem should be true for all triangulated functors?.
On the level of the DG enhancements, it is true for all smooth proper $K$-schemes that, in the homotopy category of DG categories, every functor corresponds bijectively to an isomorphism class of objects on $D(X \times Y)$. See (Toen 2006).
Lutz Hille, Michel van den Bergh, Fourier-Mukai transforms (arXiv)
Alexei Bondal, Michel van den Bergh. Generators and representability of functors in commutative and noncommutative geometry, 2002, arXiv
Banerjee and Hudson have defined Fourier-Mukai functors analogously on algebraic cobordism.
Discussion of internal homs of dg-categories in terms of refined Fourier-Mukai transforms is in
Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667
Alberto Canonaco, Paolo Stellari, Internal Homs via extensions of dg functors (arXiv:1312.5619)
For a discussion of Fourier-Mukai transforms in the setting of $(\infty,1)$-enhancements, see