Title: Lie algebroids are the same as involution algebroids in the category of smooth manifolds.
Abstract: Involution algebroids are a generalisation of Lie algebroids that make sense in any tangent category. The aim of this talk is to sketch a proof that the category of Lie algebroids is isomorphic to the category of involution algebroids in the category of smooth manifolds. Our method is to use the structure equations of the Lie algebroid to mediate between the two definitions. The advantage of this approach is that it reveals that the Tulczyjew involution in Lagrangian mechanics satisfies the involution algebroid axioms. This is joint work with Ben MacAdam and is based on an idea of Richard Garner.