# Calabi-Yau structures on dg-categories

A (smooth) dg algebra is called (left) Calabi-Yau if (see for example here) $$A^! = A[-n]$$ Here we use the inverse dualizing complex $$A^!=\mathbf{R}\operatorname{Hom}_{(A^e)^{op}}(A,A^e)$$. In topological string theory (e.g. in the context of matrix factorizations, see here or here), it is interesting to regard $$A$$ as a dg-category and form a sort of Yondeda-embedding $$\hat{A} := \operatorname{Int}(A-{\operatorname{Mod}})$$ defined in Toen of it, the full subcategory of fibrant-cofibrant objects in the category of $$A$$-dg-modules. 9 My questions are the following:

• Is the dg-category $$\hat{A}$$ (where the dg-structure is inherited from $$A-{\operatorname{Mod}}$$ automatically a Calabi-Yau $$A_\infty$$-category when regarded as an $$A_\infty$$-category? So, are both of these definitions of Calabi-Yau compatible? Due to the work of Costello, that would be required for the above construction to actually define a TCFT, and it is often stated it actually does - however I have never actually seen a discussion of this subtle point.
• If this is not always the case, is the homotopy category of $$\hat{A}$$ at least a Calabi-Yau category?