Categorical structures of Kuranishi spaces with $L_{\mathrm{\infty }}[1]$-algebras

We introduce $L_{\mathrm{\infty }}$-Kuranishi spaces by associating, to each chart, $L_{\mathrm{\infty }}[1]$-algebras defined on open neighborhoods of points in the zero locus of the Kuranishi section. We show that these objects collectively form a category into which the category of smooth manifolds naturally embeds. Some notions in \cite{FOOO1} are modified to achieve the desired categorical structures; for instance, the tangent bundle condition for chart embeddings is replaced by a quasi-isomorphism condition for the $L_{\mathrm{\infty }}[1]$-structures.