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.