Relative Calabi-Yau completions

Abstract

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 $\tilde{\eta }\in H{N}_{n-2}(B,A)$. We prove that these extensions admit relative $n$-Calabi-Yau structures in the sense of \cite{BD19}. In particular, this proof covers the original absolute case of \cite{Kel11}.

33 pages. Comments welcome

References in corpus (4)

• Higher Topos Theory
• Calabi-Yau algebras
• On the perturbation lemma, and deformations
• Geometry of Multiplicative Preprojective Algebra

Cited by in corpus (3)

• Pre-Calabi-Yau structures and moduli of representations
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Erratum to "Deformed Calabi-Yau completions"