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

TL;DR

The paper develops a structure theory for étale fibrations of $L_infty$-bundles in the $C^ine$-context, proving that linear étale fibrations admit local sections built from simple, transfer-embedded pieces. It then derives an inverse function theorem for $L_infty$-bundles and shows that weak equivalences induce quasi-isomorphisms on the global sections’ dg-algebras, enabling a streamlined description of the homotopy category via germ morphisms and path-space homotopies. By combining transfer techniques with splitting and Koszul-type resolutions, the authors provide a concrete toolkit for analyzing morphisms, fibrations, and quasi-isomorphisms in derived differential geometry. Applications include a transparent description of the homotopy category of $L_infty$-bundles and a practical criterion linking geometric weak equivalences to algebraic quasi-isomorphisms of global function algebras.

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.

Theorems & Definitions (34)

• Proposition 1.1: BLX1
• Theorem 1.2: BLX1
• Remark 1.3
• Proposition 1.4: Homotopy Transfer Theorem
• Remark 1.5
• Proposition 1.6
• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4