ah, am just rediscovering this thread here, after over 3.5 years.

The answer to #2 is of course: the $A_\infty$-category of coherent sheaves on a smooth projective CY variety is a CY-$A_\infty$-category. See here.

]]>Right, that’s the natural next question.

I am still in a rush and only had a handful of minutes. But as a start I have:

added the example of CY-objects that are CY-algebras to the CY-object entry

created an entry on Calabi-Yau algebra with a brief indication of the relation to CY-manifolds.

Not much so far, but I give some pointers to the literature. Have to dash off now…

]]>Is there a sense in which Calabi-Yau object is a generalization of Calabi-Yau manifold (for some definition…maybe trivial canonical bundle and some vanishing cohomology conditions)? I guess since this definition is for symmetric monoidal (infinity, 2)-categories, the question should be rephrased: is there a way to consider a Calabi-Yau manifold $X$ as an object in $\mathrm{Bord}_2$ such that this is a Calabi-Yau object?

]]>started Calabi-Yau object. But am being interrupted…

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