The free tangent structure

At the heart of differential geometry is the construction of the tangent bundle of a manifold. There are various abstractions of this construction, and this paper seeks to compare two of them: Synthetic Differential Geometry (SDG) and Tangent Structures. Tangent structure is defined via giving an underlying category M and a tangent functor T along with a list of natural transformations satisfying a set of axioms, then detailing the behaviour of T in the category End(M). SDG on the other hand is defined through the use of Weil algebras. The aim of this paper is to present a more precise relationship between the two approaches for describing tangent structures.