Homotopy theory of pre-Calabi-Yau morphisms

In this article we study the homotopy theory of pre-Calabi-Yau morphisms, viewing them as Maurer-Cartan elements of an L∞-algebra. We give two different notions of homotopy: a notion of weak homotopy for morphisms between d-pre-Calabi-Yau categories whose underlying graded quivers on the domain (resp. codomain) are the same, and a notion of homotopy for morphisms between fixed pre-Calabi-Yau categories (A,sd+1MA) and (B,sd+1MB). Then, we show that the notion of homotopy is stable under composition and that homotopy equivalences are quasi-isomorphisms. Finally, we prove that the functor constructed by the author in a previous article between the category of pre-Calabi-Yau categories and the partial category of A∞-categories of the form A^*[d-1], for A a graded quiver, together with hat morphisms sends homotopic d-pre-Calabi-Yau morphisms to weak homotopic A∞-morphisms.