Synthetic differential geometry

Definition

In mathematics, synthetic differential geometry is a formalization of the theory 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. 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.

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Abraham Robinson Adequality Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes Augustin-Louis Cauchy Category theory Cavalieri's principle Constructive nonstandard analysis Cours d'analyse Criticism of nonstandard analysis Differential (mathematics)Differential geometry Dual number Dual numbers Fibre bundle Functor Gottfried Wilhelm Leibniz Hyperfinite set Hyperinteger Hyperreal number Increment theorem Infinitesimal Infinitesimal strain theory Integral symbol Internal set Internal set theory Jet (mathematics)Jet bundle John Lane Bell Law of continuity Leibniz's notation Leonhard Euler Levi-Civita field Mathematics Michael Shulman (mathematician)Microcontinuity Monad (nonstandard analysis)Nonstandard analysis Nonstandard calculus Overspill Pierre de Fermat Representable functor Smooth infinitesimal analysis Smooth manifold Standard part function Surreal number The Analyst The Method of Mechanical Theorems Topos theory Transcendental law of homogeneity Transfer principle William Lawvere

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