Kuranishi chart categories and higher cocycle conditions

Abstract

Given an $L_\infty$-Kuranishi space introduced in \cite{Kim1}, we propose a notion called the Kuranishi chart category. Using the nerve of this category, together with a choice of atlas and a simplicial description of the covering of the underlying topological space, we formulate a higher homotopical version of the bundle-component cocycle condition. We show that this condition is always satisfied, by virtue of a property of the higher homotopy theory of $L_\infty[1]$-morphisms developed in \cite{Kim2}, concerning quasi-isomorphisms. As a consequence, the rigid cocycle condition of Fukaya-Oh-Ohta-Ono Kuranishi spaces is replaced by more flexible, homotopy-theoretic compatibility.