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 $L_{\infty}$-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 $(\mathcal{A},s_{d+1}M_{\mathcal{A}})$ and $(\mathcal{B},s_{d+1}M_{\mathcal{B}})$. 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 $A_{\infty}$-categories of the form $\mathcal{A}\oplus\mathcal{A}^*[d-1]$, for $\mathcal{A}$ a graded quiver, together with hat morphisms sends homotopic $d$-pre-Calabi-Yau morphisms to weak homotopic $A_{\infty}$-morphisms.