12.2 Synthetic differential geometry

Synthetic differential geometry revolutionizes calculus by using infinitesimals instead of limits. This approach simplifies complex calculations and provides a more intuitive foundation for advanced mathematical concepts.

The theory's principles, like microlinearity and integration axioms, allow for elegant formulations of geometric ideas. Models in smooth topoi provide consistent interpretations, bridging the gap between intuition and rigorous mathematics.

Foundations of Synthetic Differential Geometry

Principles of synthetic differential geometry

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• Infinitesimal elements form basis of calculus without limits
  • Nilpotent infinitesimals square to zero ($d^2 = 0$)
  • Kock-Lawvere axiom defines unique solutions for certain polynomial equations
• Smooth functions generalize differentiable functions
  • Microlinearity ensures local linearity at infinitesimal scales
  • Infinitesimal linearity allows Taylor expansions with nilpotent terms
• Axiom of integration enables definition of integral without limits
• Well-adapted models provide consistent interpretations (topos theory)
• Principle of microaffinity guarantees existence of tangent spaces
• Principle of microcancellation allows cancellation of infinitesimals in equations

Models in suitable topoi

• Smooth topoi provide framework for synthetic differential geometry
  • Cahiers topos uses sheaves on infinitesimal sites
  • Dubuc topos employs C∞-rings and ideals
• Germ-determined functions capture local behavior near a point
• Smooth loci generalize notion of smooth manifolds
• Grothendieck topologies define sheaves on categories
• Sheaf models interpret synthetic differential geometry axioms
• Presheaf categories offer flexible settings for constructing models

Applications and Comparisons

Applications to analysis and geometry

• Tangent vectors and bundles defined using infinitesimal paths
• Differential forms represent infinitesimal areas and volumes
• De Rham cohomology computed using nilpotent infinitesimals
• Lie groups and algebras studied with infinitesimal symmetries
• Manifolds in synthetic context generalize smooth spaces
• Calculus of variations solved using infinitesimal perturbations
• Symplectic geometry formulated with infinitesimal phase spaces

Synthetic vs classical differential geometry

• Intuitionistic logic allows infinitesimals, classical logic doesn't
• Smooth infinitesimal analysis provides rigorous foundation
• Treatment of singularities more natural in synthetic approach
• Coordinate-free approach emphasizes intrinsic geometry
• Generalization of manifolds includes singular and infinite-dimensional spaces
• Nilpotent infinitesimals simplify calculations (derivatives, Taylor series)
• Consistency with classical results ensured by transfer principle
• Advantages in certain geometrical constructions (tangent bundles, jet spaces)