$L_{\mathrm{\infty }}$-Kuranishi spaces and the moduli space of pseudoholomorphic maps

We introduce $L_{\mathrm{\infty }}$-Kuranishi spaces by associating, to each chart, $L_{\mathrm{\infty }}[1]$-algebras defined on open neighborhoods of the zero points of the Kuranishi section. We show that these objects collectively form a category, which naturally embeds the category of smooth manifolds. Certain notions in \cite{FOOO1} are modified to achieve desired categorical structures; for instance, the tangent bundle condition is interpreted as a quasi-isomorphism condition for the $L_{\mathrm{\infty }}$-structures. In this process, the originally strict and rigid cocycle condition for coordinate changes is replaced by more flexible homotopy-theoretic compatibilities. To this end, a model of higher homotopy theory for $L_{\mathrm{\infty }}[1]$-morphisms is proposed. Moreover, the moduli space of pseudoholomorphic disks with Lagrangian boundary condition is shown to serve as an example of $L_{\mathrm{\infty }}$-Kuranishi spaces, provided that a Whitney stratification with a compatible system of tubular neighborhoods exists on each chart. Finally, the forgetful and evaluation maps for the moduli space are lifted to morphisms between $L_{\mathrm{\infty }}$-Kuranishi spaces.