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{{Short description|A formalization of the theory of differential geometry in the language of topos theory}} |
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{{Short description|Formalization in mathematical topos theory}} |
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{{inline|date=November 2011}} |
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{{inline|date=November 2011}} |
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In ], '''synthetic differential geometry''' is a formalization of the theory of ] in the language of ]. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of ]s can be encoded into certain ]s on manifolds: namely bundles of ] (see also ]). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is ] in nature. The third insight is that over a certain ], these are ]s. Furthermore, their representatives are related to the algebras of ], so that ] may be used. |
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In ], '''synthetic differential geometry''' is a formalization of the theory of ] in the language of ]. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of ]s can be encoded into certain ]s on manifolds: namely bundles of ] (see also ]). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is ] in nature. The third insight is that over a certain ], these are ]s. Furthermore, their representatives are related to the algebras of ], so that ] may be used. |
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== Further reading == |
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== Further reading == |
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*], (PDF file) |
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*], (PDF file) |
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*], (PDF file) |
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*], (PDF file) |
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*Anders Kock, (PDF file), Cambridge University Press, 2nd Edition, 2006. |
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*Anders Kock, (PDF file), Cambridge University Press, 2nd Edition, 2006. |
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*R. Lavendhomme, ''Basic Concepts of Synthetic Differential Geometry'', Springer-Verlag, 1996. |
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*R. Lavendhomme, ''Basic Concepts of Synthetic Differential Geometry'', Springer-Verlag, 1996. |
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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.
