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
The specialization of a context of six operations $(f^\ast \dashv f_\ast)$, $(f_! \dashv f^!)$ to the case that the “projection formula” $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$ holds naturally in $X,Y$. (May 05, def. 2.12)
In a Verdier-Grothendieck context, duality intertwines $f_!$ with $f_\ast$ and $f^!$ with $f^\ast$.
(May 05, section 3, Joshua, corollary 5.4)
A general abstract discussion of the axioms and their consequences is in
A fairly general class of implementations is in