The inverse function theorem for curved $L$-infinity spaces

Abstract

In this paper, we prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved $L_{\infty }$ 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 $L_{\infty }$ algebras. The main ingredients are the obstruction theory for $L_{\infty }$ homomorphisms (in the curved setting) and the homotopy transfer theorem for curved $L_{\infty }$ algebras. Both techniques work in the $A_{\infty }$ case as well.