Product-preserving functors on smooth manifolds (Q5903285)

A complete classification is given of functors F from the category of \(C^{\infty}\) manifolds to itself that preserve products - F(M\(\times N)\) is naturally equivalent to F(M)\(\times F(N)\)- and embeddings. The primary examples of such functors are the k-jet functors, \((\cdot)^ n_ k\), defined by \(M^ k_ h=\{k\)-jets at 0 of \(C^{\infty}\) maps \({\mathbb{R}}^ n\to M\}\). On the category of connected manifolds, any product-preserving functor can be obtained by taking a quotient of products of such k-jet bundles. In fact, there is a natural equivalence of categories between the category of product-preserving functors on connected manifolds and the category of finite-dimensional \({\mathbb{R}}\)- algebras all of whose quotient fields are \({\mathbb{R}}\). The algebra corresponding to a functor F is just F(\({\mathbb{R}})\), and the algebra corresponding to \((\cdot)^ n_ k\) is \({\mathbb{R}}[x_ 1,...,x_ n]/(x_ 1,...,x_ n)^{k+1}\). The classification in the case of connected manifolds can also be described as follows: A functor F can be thought of as corresponding to an ''infinitesimal manifold'' whose ring of \(C^{\infty}\) \({\mathbb{R}}\)-valued functions is F(\({\mathbb{R}})\). Such a ''manifold'' has a finite number of points, but each point can have a nontrivial infinitesimal neighborhood. F(M) can be obtained as the set of maps from this manifold to M. This classification extends nicely to the full category of manifolds: Given an infinitesimal manifold, divide its points into equivalence classes. Let F(M) be the set of maps from this manifold to M which respect equivalence classes in the sense that all the points in a given equivalence class are mapped to the same connected component of M. The result is a product-preserving functor, and all such functors are obtained in this way.