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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarted [[Calabi-Yau object]]. But am being interrupted...

   started Calabi-Yau object. But am being interrupted…

2. • CommentRowNumber2.
   • CommentAuthorhilbertthm90
   • CommentTimeMay 28th 2011
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   Author: hilbertthm90
   Format: MarkdownItexIs 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 29th 2011
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   Author: Urs
   Format: MarkdownItexRight, 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 <a href="http://ncatlab.org/nlab/show/Calabi-Yau+object#CYAlgebra">example of CY-objects that are CY-algebras</a> 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...

   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…

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 18th 2014
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   Author: Urs
   Format: MarkdownItexah, 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](http://ncatlab.org/nlab/show/Calabi-Yau+category#FromCYVarieties).

   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.