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Geometry and Topology
Buy Now. Add to my Library. Synopsis Starting at an introductory level, this work leads to important and often new results in synthetic differential geometry. From rudimentary analysis the book moves to such results as: a new proof of De Rham's theorem; the synthetic view of global action, going as far as the Weil characteristic homomorphism; the systematic account of structured Lie objects, such as Riemannian, symplectic, or Poisson Lie objects; the view of global Lie algebras as Lie algebras of a Lie group in the synthetic sense; and, lastly, the synthetic construction of symplectic structure on the cotangent bundle in general.
Thus, while the book is limited to a naive point of view developing synthetic differential geometry as a theory in itself, the author nevertheless treats somewhat advanced topics, which are classic in classical differential geometry but new in the synthetic context. Shipping This title is in stock with our Australian supplier and arrives at our Sydney warehouse within weeks of your order. Cartan and their contemporaries explicitly used infinitesimal real numbers, infinitesimal curves, etc.
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In present day mathematics, based on set-theoretic foundations, such infinitesimal reals do not exist, and the synthetic methods cannot be made mathematically rigorous in a direct way. The phrase "synthetic differential geometry" usually refers to a development initiated by F.
Lawvere's lecture later published as [a2]. This development is based on category-theoretic rather than set-theoretic foundations, and is compatible with infinitesimals. For example, one of the basic axioms often called the Kock—Lawvere axiom states that for the subobject of the line the mapping , sending to the "infinitesimal" straight line , is an isomorphism.
Basic concepts of synthetic differential geometry
Thus, the derivative of a function at a point can be defined in a purely algebraic way as the unique number for which for all ; no limits are involved in this definition. Much of "infinitesimal" differential geometry, such as the theory of connections, curvature, etc. It is remarkable that in the context of synthetic differential geometry infinitesimal arguments such as those by Cartan literally make sense, and are mathematically rigorous.
The relation with ordinary differential geometry is established via suitable models, as presented in [a3] and [a4]. These models are Grothendieck topoi cf. Topos , much like the ones in algebraic geometry, but based on rings of smooth functions on manifolds, and quotients of such rings by ordinary ideals.
Following Lawvere, every topos can be viewed as a universe of sets, with an intrinsic logic which is intuitionistic.
Thus, it is possible to give foundations of synthetic differential geometry which are based on intuitionistic set theory rather than on category theory this is done in [a5]. It should be emphasized that the infinitesimals used in synthetic differential geometry are generally nilpotent, and hence cannot be accounted for in Robinson's non-standard analysis.
The compatibility of non-standard analysis with synthetic differential geometry is demonstrated in [a4]. For detailed expositions of various aspects of synthetic differential geometry see [a5] , [a7] , [a8]. Log in.
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