Functoriality of the cotangent bundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T16:42:28Z http://mathoverflow.net/feeds/question/62356 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62356/functoriality-of-the-cotangent-bundle Functoriality of the cotangent bundle Theo Johnson-Freyd 2011-04-20T00:47:13Z 2011-04-20T00:47:13Z <p>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).</p> <p>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.</p> <p>My question is:</p> <blockquote> <p>Do the étale maps comprise the largest class of morphisms of manifolds for which $\mathrm T^*$ is functorial?</p> </blockquote> <p>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.</p>