Homotopy Transfer Theorem and minimal models for pre-Calabi-Yau categories

In this article, we extend results of J. Leray and B. Vallette on homotopical properties of pre-Calabi-Yau algebras to the case of pre-Calabi-Yau categories. We give direct proofs of the results adapting techniques used by D. Petersen for the case of coalgebras, instead of using properadic calculus. More precisely, we prove that given quasi-isomorphic dg quivers and a pre-Calabi-Yau structure on one of them there exists a pre-Calabi-Yau structure on the other as well as a pre-Calabi-Yau morphism between the two. Moreover, we show that a pre-Calabi-Yau morphism whose first component is an isomorphism of dg quivers is an isomorphism in the category of pre-Calabi-Yau categories. We incidentally show that any pre-Calabi-Yau category has a minimal model. Finally, we prove that any quasi-isomorphism of pre-Calabi-Yau categories admits a quasi-inverse.