In
mathematics, synthetic differential geometry is a formalization of the theory of
differential geometry in the language of
topos theory
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
. 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 manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s can be encoded into certain
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
s on manifolds: namely bundles of
jets (see also
jet bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...
). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is
functorial
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
in nature. The third insight is that over a certain
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
, these are
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets a ...
s. Furthermore, their representatives are related to the algebras of
dual numbers
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
, 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.
Further reading
*
John Lane Bell
John Lane Bell (born March 25, 1945) is an Anglo-Canadian philosopher, mathematician and logician. He is Professor Emeritus of Philosophy at the University of Western Ontario in Canada. His research includes such topics as set theory, model theor ...
Two 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