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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 ?".

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This article of Fausk, Hu and May adds nothing to the list but gives a very nice analysis of the categorical situations in which you have six operations.

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