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$L_{\infty }$ SPACES
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.
