nLab vertical tangent Lie algebroid

For $\pi : Y \to X$ a morphism of smooth manifolds, the vertical tangent Lie algebroid with respect to $\pi$ is the sub-Lie algebroid $T_{vert} Y \hookrightarrow T X$ of the tangent Lie algebroid $T Y$ of $Y$ whose Chevalley-Eilenberg algebra is the dg-algebra of horizontal differential forms on $Y$ with respect to $\pi$.

For more details on the construction see the examples section at exterior differential systems.

The vertical tangent Lie algebroid is the infinitesimal version of the vertical path ∞-groupoid. It plays a central role in the context of Ehresmann connections and Cartan-Ehresmann ∞-connections.