From $L_{\mathrm{\infty }}$ algebroids to $L_{\mathrm{\infty }}$ spaces: Part I

The notion of $L_{\mathrm{\infty }}$ spaces over dg manifolds is developed. An equivalence between the category of transitive $L_{\mathrm{\infty }}$ algebroids and that of $L_{\mathrm{\infty }}$ spaces is established, and this equivalence detects weak equivalences. Moreover, a faithful functor from $L_{\mathrm{\infty }}$ algebroids to $L_{\mathrm{\infty }}$ spaces is constructed, which also detects weak equivalences.