On the structure of étale fibrations of $L_{\infty }$-bundles

Abstract

We prove that an étale fibration between $L_{\infty }$-bundles admits local sections composed of several elementary morphisms of particularly simple and accessible type. As applications, we establish an inverse function theorem for $L_{\infty }$-bundles and provide an elementary proof that every weak equivalence of $L_{\infty }$-bundles induces a quasi-isomorphism of the differential graded algebras of global functions. Furthermore, we apply this inverse function theorem to show that the homotopy category of $L_{\infty }$-bundles admits a simple description in terms of homotopy classes of morphisms, when $L_{\infty }$-bundles are restricted to their germs around their classical loci.

24 pages. minor revision. arXiv admin note: significant text overlap with a part of arXiv:2006.01376