$A_{\mathrm{\infty }}$-Deformations and their Derived Categories

Handling curved $A_{\mathrm{\infty }}$-deformations is challenging and defining their derived categories seems impossible. In this paper, we show how to welcome the curvature and build derived categories despite the apparent difficulties. We explicitly describe their $A_{\mathrm{\infty }}$-structure by a deformed Kadeishvili theorem. This paper starts with a review of $A_{\mathrm{\infty }}$-deformations for non-specialists. We explain why $A_{\mathrm{\infty }}$-deformations require curvature and why curvature supposedly causes problems. We prove that infinitesimally curved $A_{\mathrm{\infty }}$-deformations are in fact non-problematic and provide the correct definitions for their twisted completion and minimal model. The classical Kadeishvili theorem collapses for curved deformations. We show how to adapt it by iteratively optimizing the curvature and changing the homological splitting infinitesimally. The final $A_{\mathrm{\infty }}$-structure on the minimal model is constructed in terms of trees.