- Synthetic differential geometry
In

mathematics ,**synthetic differential geometry**is a reformulation ofdifferential geometry in the language oftopos theory . There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class ofsmooth manifold s can be encoded into certainfibre bundle s on manifolds: namely bundles of jets (see alsojet 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 arerepresentable functor s. Furthermore, their representatives are related to the algebras ofdual numbers , so thatsmooth 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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