We show in this paper that the correspondence between $2$-termrepresentations up to homotopy and $\mathcal{VB}$-algebroids, established byGracia-Saz and Mehta, holds also at the level of morphisms. This correspondenceis hence an equivalence of categories. As an application, we study foliationsand distributions on a Lie algebroid, that are compatible both with the linearstructure and the Lie algebroid structure. In particular, we show howinfinitesimal ideal systems in a Lie algebroid $A$ are related withsubrepresentations of the adjoint representation of $A$.
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机译：我们在本文中证明，由Gracia-Saz和Mehta建立的$ 2 $-直到同态的术语表示与$ \ mathcal {VB} $-代数之间的对应关系也保持在射态水平。因此，这种对应是类别的等价。作为一种应用，我们研究了与线性结构和李代数结构都兼容的李代数上的叶面和分布。特别是，我们显示了李代数$ A $中的极小理想系统与$ A $伴随表示的子表示相关。
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