In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory.
The second insight is that the operation of assigning a bundle of jets to a smooth manifold is functorial in nature.
Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used.
Synthetic differential geometry can serve as a platform for formulating certain otherwise obscure or confusing notions from differential geometry.
For example, the meaning of what it means to be natural (or invariant) has a particularly simple expression, even though the formulation in classical differential geometry may be quite difficult.