Synthetic differential geometry

In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle). A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. In Mathematics, the jet is an operation which takes a Differentiable function f and produces a Polynomial, the truncated Taylor polynomial In Differential geometry, the jet bundle is a certain construction which makes a new smooth Fiber bundle out of a given smooth fiber bundle The second insight is that the operation of assigning a bundle of jets to a smooth manifold is functorial in nature. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories The third insight is that over a certain category, these are representable functors. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, especially in Category theory, a representable functor is a Functor of a special form from an arbitrary category into the Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used. A variety of dualities in mathematics are listed at Duality (mathematics. Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of Infinitesimals Based on the ideas of F

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.

Further reading