- Synthetic differential geometry
mathematics, synthetic differential geometry is a reformulation of differential geometryin 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 analysismay 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.
*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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