Recall that to any manifold $X$, I can assign in a canonical way a manifold $\mathrm T ^* X$, the total space of the cotangent bundle over $X$. Recall also that, unlike the tangent bundle construction, the map $X \mapsto \mathrm T^* X$ is not an endofunctor on the category of manifolds: whereas tangent vectors push forward along smooth maps, cotangent (co)vectors do not (they also do not pull back).
Nevertheless, $X \mapsto \mathrm T^* X$ is functorial for some restricted classes of maps. For example, there is a category whose objects are manifolds and whose morphisms are étale maps, and the cotangent construction is (covariantly) functorial for this category.
My question is:
Do the étale maps comprise the largest class of morphisms of manifolds for which $\mathrm T^*$ is functorial?
In my particular situation, I have a (surjective) submersion $Y \to X$, and I can construct by hand a (Poisson) map $\mathrm T^* Y \to \mathrm T^*X$ covering it, because I know of some extra structure for $Y,X$. But I would like to know if there is a more canonical reason that I have this map.
Best Answer
I do not know a larger class of smooth mappings; and I considered this question intensively when co-writing the book "Natural operations in differential geometry, Springer-Verlag, 1993"(pdf).
See also 26.11 -- 26.16 in this book for a determination of all natural transformations $T T^* \to T^*T$, viewed as functors on the category of $m$-dimensional manifolds and local diffeomorphisms (etale mappings), and similar questions.