Minimal $A_{\mathrm{\infty }}$-algebras of endomorphisms: The case of $d\mathbb{Z}$-cluster tilting objects

The Derived Auslander–Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic $d\mathbb{Z}$-cluster tilting objects in $\mathrm{Hom}$-finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal $A_{\mathrm{\infty }}$-algebra structures in the proof of this result, as well as the crucial role of the enhanced $A_{\mathrm{\infty }}$-obstruction theory developed by the second-named author.