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Abstract

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty}$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty}$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty}$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty}$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty}$ space.