# When do six operations work?

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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## 1 Answer

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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