Differential Structure, Tangent Structure, and SDG

In 1984, J. Rosický gave an abstract presentation of the structure associated to tangent bundle functors in differential and algebraic geometry. By slightly generalizing this notion, we show that tangent structure is also fundamentally related to the more recently introduced Cartesian differential categories. In particular, tangent structure of a trivial bundle is precisely the same as Cartesian differential structure. We also provide a general result which shows how tangent structure arises from the manifold completion (in the sense of M. Grandis) of a differential restriction category. This construction includes all standard atlas-based constructions from differential geometry. Furthermore, we tighten the relationship, which Rosický had noted, between representable tangent structure and synthetic differential geometry, showing how such settings can be developed from a system of infinitesimal objects. We also show how infinitesimal objects give rise to dual tangent structure. Taken together, these results show that tangent structures appropriately span a very wide range of definitions, from the syntactic and structural differentials arising in computer science and combinatorics, through the concrete manifolds of algebraic and differential geometry, and finally to the abstract definitions of synthetic differential geometry.