nLab Lie algebroid-groupoid

Idea

An internal groupoid in the category of Lie algebroids.

Examples

For $\mathcal{G}_\bullet$ a Lie groupoid, forming degreewise the tangent Lie algebroid yields the tangent Lie algebroid groupoid

$$\array{ T \mathcal{G}_1 &\to& \mathcal{G}_1 \\ {}^{\mathllap{d s}}\downarrow \downarrow^{\mathrlap{d t}} && {}^{\mathllap{s}}\downarrow \downarrow^{\mathrlap{t}} \\ T \mathcal{G}_0 &\to& \mathcal{G}_0 } \,.$$

Related concepts

• double Lie algebroid, Lie algebroid-groupoid, double Lie groupoid

• foliation of a Lie groupoid

References

The notion is considered (under the name “$\mathcal{L A}$-groupoids”) in

• Kirill Mackenzie, Double Lie algebroids and second-order geometry. I. Adv. Math. 94 (1992), no. 2, 180–239

• Kirill Mackenzie, Double Lie algebroids and second-order geometry. II. Adv.Math. 154 (2000), no. 1, 46–75. (arXiv:dg-ga/9712013).