This question comes (heavily edited) from my notes, thus slightly unusual structure.
We know that algebraic maps have very strict structure, and in many settings the operations f_*, f_!, their adjoints f^*, f^!, bioperations ⊗ and => as well as duality D behave well. They satisfy (whenever defined) some good identities, especially for proper morphisms.
There are specific subtleties in the following cases:
case Z: constructible sheaves
:=(finite) local systems (finitely) glued ...case O: coherent sheaves
:=finitely generated O-modules ...case D: (D-modules) holonomic
:='number of equations is just right' ...
Question: I wonder if there are other sheaves of non-commutative algebras for which we can define operations and duality? That is, is it possible to continue this list with another "case ?".
