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Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.

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How is this action of monoidal derived category induced?

The original action takes the form of an additive functor $A \times B \to B$ with notation as in the question (and appropriate coherence conditions giving compatibility with the monoidal structure on …
Greg Stevenson's user avatar