The inverse function theorem for curved L-infinity spaces

In this paper we prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved $L_{\mathrm{\infty }}$ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. This theorem simultaneously generalizes the inverse function theorem for smooth manifolds and the Whitehead theorem for $L_{\mathrm{\infty }}$ algebras. The main ingredients are the obstruction theory for $L_{\mathrm{\infty }}$ homomorphisms (in the curved setting) and the homotopy transfer theorem for curved $L_{\mathrm{\infty }}$ algebras. Both techniques work in the $A_{\mathrm{\infty }}$ case as well.