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 ∞ 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 ∞ algebras. The main ingredients are the obstruction theory for L ∞ homomorphisms (in the curved setting) and the homotopy transfer theorem for curved L ∞ algebras. Both techniques work in the A ∞ case as well.