Denote by \({\mathcal M}\) the category of smooth manifolds and by \(T: {\mathcal M}\to {\mathcal M}\) the tangent functor. The tangent functor category (t.f.c.) has as objects the (iterated) tangent functors \(T^ n: {\mathcal M}\to {\mathcal M}\) (n\(\geq 0)\) and as morphisms the natural transformations. It is known that t.f.c. is the same as the category with objects \(T^ n: {\mathcal M}\to Sets\) (i.e. to each manifold there corresponds the set of points of its n-tangent bundle) and morphisms the natural transformations.
In this paper it is shown that t.f.c. can equivalently be described by enlarging \({\mathcal M}\) to the opposite category \({\mathcal F}\) of the category of finitely presented \(C^{\infty}\)-rings. \({\mathcal F}\) may be thought of as a finite inverse limit completion of \({\mathcal M}\). Under this process, \({\mathcal F}\) contains a particular object D such that, roughly speaking, maps of the form \(D\to M\) in \({\mathcal F}\) correspond to points on TM, each \(T^ n: {\mathcal M}\to Sets\) extends to \(Hom_{{\mathcal F}}(D^ n\),-): \({\mathcal F}\to Sets\) and a natural transformation corresponds uniquely to an \({\mathcal F}\)-map \(D^ m\to D^ n\) between “infinitesimal manifolds”. Hence, t.f.c. can equivalently be described as the dual of the full subcategory whose objects are the “infinitesimal manifolds” \(D^ m\). Each \(T^ n\) corresponds to \(D^ n\) and natural transformations correspond to \({\mathcal F}\)-maps \(D^ m\to D^ n\). This allows the authors to derive a number of (already known) results concerning t.f.c. in a simple and elegant way.