In the paper Integrability of Lie Brackets Marius Crainic and Rui Fernandes describe obstructions to integrate a Lie algebroid to a Lie groupoid. The process of integration relies on the construction of the **Weinstein groupoid** which is the quotient of the space of A-paths by a suitable equivalence relation.

Given a Lie algebroid $A\stackrel{\pi}{\longrightarrow} M$, an $A$-path is a path $a:I\longrightarrow A$ of class $C^1$ such that $$\sharp a(t)=\frac{d}{dt}\pi(a(t))$$ where $\sharp:A\longrightarrow TM$ is the anchor map.

The RHS of the above equality (I believe) stands for the map $$t\longmapsto ((\pi\circ a)(t), (\pi\circ a)^\prime(t)).$$

They state:

An $A$-path can be seen as a morphism of vector bundles $a\ dt:TI\longrightarrow A$ covering $\pi\circ a:I\longrightarrow M$ and this gives a Lie algebroid morphism $TI\longrightarrow A$.

I don't understand this statement.

Firstly, what does $dt$ in $a\ dt$ mean? Is it just a notation or what?

Furthermore, can anyone explain-me with details the equivalence between A-paths and Lie algebroid morphisms $TI\longrightarrow A$?

1more comment