Do Lie algebroids pull back (along submersions)?

Question

There are more general definitions, but for my purposes a Lie algebroid on a smooth manifold $X$ is a vector bundle $A \to X$, a map $\rho: A \to {\rm T}X$ of vector bundles over $X$, and a bracket $[,]$ making $\Gamma(A)$ into an $\mathbb R$-Lie algebra, such that $\rho$ induces is a map $\Gamma(A) \to \Gamma({\rm T}X)$ of Lie algebras and such that the Leibniz rule $[a,fb] = f[a,b] + (\rho(a)f)b$ is satisfied for $f\in \mathcal C^\infty$ and $a,b\in \Gamma(A)$.

So, suppose I happen to have a Lie algebroid $A\to X$ lying around, and also a smooth map $\phi: Y\to X$. (In my case, $Y \to X$ happens to be a vector bundle, so you can assume some fairly strong properties of the map.) Then I can certainly pull back the vector bundle $A\to X$ to $\phi^*A \to Y$. Is there a natural Lie algebroid structure I can put on this pullback? The answer is probably "how natural do you want it?": if $Y = \{{\rm pt}\}$ and $\phi({\rm pt}) = y\in X$, then the anchor map for $\phi^*A \to \{{\rm pt}\}$ must be trivial, but the only part of the fiber $A_y$ with a Lie algebra structure is $\ker \{A_y \overset\rho\to {\rm T}_y X\}$.

So the real question is:

Along what type of maps do Lie algebroids pull back? In particular, can I always pull back a Lie algebroid along a submersion?

Answer 1

As your example suggests, the vector bundle pullback is perhaps not the right thing to consider when wanting defining the notion of a pullback Lie algebroid. You have to go one step further and take the pullback of $\phi^*\rho: \phi^* A \to \phi^*TX$ along $d\phi: TY \to \phi^*TX$. The resulting bundle $\phi^{**}A$ (which exists when $d\phi - \phi^*\rho: TY \oplus \phi^*A \to \phi^*TX$ has constant rank, so in particular when $\phi$ is a submersion) then has a Lie algebroid structure with gives the right universal property you would want, if that's what you want. In the case where $Y$ is a point, this gives exactly your example.

This is all detailed in the paper "Algebraic Constructions in the Category of Lie Algebroids" by Higgins and Mackenzie. This may even have something to say about just trying to do something with $\phi^*A$ itself, but I haven't checked.