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arXiv · 2307.08179

On the structure of \'etale fibrations of $L_\infty$-bundles

Kai Behrend·Hsuan-Yi Liao·Ping Xu
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Abstract

We prove that an \'etale 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.

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Kai Behrend, Hsuan-Yi Liao, Ping Xu. 2023-07-17. On the structure of \'etale fibrations of $L_\infty$-bundles. https://doi.org/10.1112/plms.70115

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