Relative Calabi-Yau completions

We generalize Keller’s construction \cite{Kel11} of deformed $n$-Calabi-Yau completions to the relative context. This gives a construction that extends any given dg functor $F:A\to B$ between smooth dg categories to a dg functor $\tilde{F}:\tilde{A}\to \tilde{B}$, together with a family of deformations of $(\tilde{A},\tilde{B},\tilde{F})$ parametrized by relative negative cyclic homology classes $\widetilde{\eta} \in HN_{n-2}(B, A)$\mathrm{.}Weprovethattheseextensionsadmitrelative$n$-Calabi-Yau structures in the sense of \cite{BD19}. In particular, this proof covers the original absolute case of \cite{Kel11}.