Synthetic differential geometry
- Synthetic differential geometry
In mathematics, synthetic differential geometry is a reformulation of differential geometry 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 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
*J.L. Bell, [http://publish.uwo.ca/~jbell/Two%20Approaches%20to%20Modelling%20the%20Universe.pdf Two Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets] (PDF file)
*F.W. Lawvere, [http://www.acsu.buffalo.edu/~wlawvere/SDG_Outline.pdf Outline of synthetic differential geometry] (PDF file)
*Anders Kock, [http://home.imf.au.dk/kock/sdg99.pdf Synthetic Differential Geometry] (PDF file), Cambridge University Press, 2nd Edition, 2006.
*R. Lavendhomme, "Basic Concepts of Synthetic Differential Geometry", Springer-Verlag, 1996.
*Michael Shulman, [http://www.math.uchicago.edu/~shulman/exposition/sdg/pizza-seminar.pdf Synthetic Differential Geometry]
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