In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, synthetic differential geometry is a formalization of the theory of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
in the language of
topos theory. 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). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is
functorial in nature. The third insight is that over a certain
category, these are
representable functors. Furthermore, their representatives are related to the algebras of
dual numbers, so that
smooth infinitesimal analysis
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being ...
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.
Further reading
*
John Lane BellTwo Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets(PDF file)
*
F.W. LawvereOutline of synthetic differential geometry(PDF file)
*Anders Kock
Synthetic Differential Geometry(PDF file), Cambridge University Press, 2nd Edition, 2006.
*R. Lavendhomme, ''Basic Concepts of Synthetic Differential Geometry'', Springer-Verlag, 1996.
*
Michael ShulmanSynthetic Differential Geometry*Ryszard Paweł Kostecki
Differential Geometry in Toposes
{{Infinitesimals
Differential geometry