# Functoriality of the cotangent bundle

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.

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Of course, there's a different target category in which cotangent manifolds are always functorial, that of symplectic varieties with morphisms given by (compositions of) Lagrangian subvarieties of the product. This is essentially why functoriality of D-modules is possible. –  Ben Webster Apr 20 '11 at 3:23
It is true that that Lagrangian is the graph of a map if and only if the map is etale. –  Ben Webster Apr 20 '11 at 3:24
@Ben: I'm very aware of the symplectic category. Unfortunately, I think for my application my target is not this category, but the category whose objects are Poisson manifolds and morphisms are Poisson maps (smooth maps $f$ for which the Poisson structures are $f$-related). That said, I'd love to hear more about functoriality of D-modules. –  Theo Johnson-Freyd Apr 20 '11 at 3:58
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". –  Peter Michor May 30 '13 at 16:52
@Peter: If you would like to leave your comment as an answer, perhaps with one or two more words, I would be happy to accept it. –  Theo Johnson-Freyd May 30 '13 at 17:41

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.