higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
A Grothendieck context is a pair of two symmetric monoidal categories $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ which are connected by an adjoint triple of functors such that the leftmost one is a closed monoidal functor.
This is the variant/special case of the yoga of six operations with two adjoint pairs $(f_! \dashv f^!)$ and $(f^\ast \dashv f_\ast)$ for $f_! \simeq f_\ast$.
(The other specialization of the six operations where $f^\ast \simeq f^!$ is called the Wirthmüller context).
The existence of the (derived) right adjoint $f^!$ to $f_\ast$ is what is called Grothendieck duality.
A homomorphism of schemes $f \;\colon\; X \longrightarrow Y$ induces an inverse image $\dashv$ direct image adjunction on the derived categories $QCoh(-)$ of quasicoherent sheaves
(all derived functors) If $f$ is a proper morphism of schemes then under mild further conditions there is a further right adjoint $f^!$
This is originally due to Grothendieck, whence the name. Refined accounts are in (Deligne 66, Verdier 68, Neeman 96).
Generalization of the pull-push adjoint triple to E-∞ geometry is in (LurieQC, prop. 2.5.12) and the projection formula for this is in (LurieProp, remark 1.3.14).
The original construction for quasicoherent sheaves on schemes is due to Alexander Grothendieck, whence the name “Grothendieck context”.
Further stream-lined accounts then appeared in
Further refinement and highlighting of the close relation to the categorical Brown representability theorem is in
Discussion of integral transforms in Grothendieck contexts is in
Generalization of the pull-push adjoint triple to E-∞ geometry is in
and the projection formula for this triple appears as remark 1.3.14 of
A clear discussion of axioms of six operations, their specialization to Grothendieck context and Wirthmüller context and their consequences is in