The construction of models for nonstandard analysis does, however, have a topos-theoretic description, using filterpower ?s.

$arithmetic of infinitesimals$

Moreover, the arithmetic of invertible infinitesimals in SDG has some unfamiliar aspects: for instance, mathematical induction is only valid for statements of a certain logical form, and the axiom of finite choice fails. By contrast, a limited sort of transfer principle relating a pair of specific models for SDG is proven in MSIA, but it applies only to statements of a certain logical form. In particular, this implies that the infinitesimal and infinitely large quantities in nonstandard analysis obey all the same rules of arithmetic and analysis as do the standard ones. to Chapter VII), âthere are some obvious differences.â The primary tool used in nonstandard analysis is a completely general transfer principle, saying that any statement in the ordinary world is also true in the nonstandard world. Such invertible infinitesimals can be applied in some of the same ways as the infinitesimals of nonstandard analysis. Nevertheless, one can construct models of synthetic differential geometry which, in addition to nilpotent infinitesimals, contain invertible infinitesimals see for instance MSIA, chapters VI and VII. The infinitesimal quantities considered there differ from the general ones in synthetic differential geometry in that they are all invertible (their inverses being âinfinitely largeâ). Comparison to infinitesimals in nonstandard analysisĪnother notion of infinitesimals has arisen in the context of nonstandard analysis. Similar infinitesimal thickenings also appear in the noncommutative geometry of Kapranov, and in the language of abelian categories of quasicoherent sheaves in the work of Lunts and Rosenberg on regular differential operators in the content of noncommutative geometry, which strongly takes into account tensor products. It is the foundation of Grothendieckâs approach to regular differential operators, to costratifications, crystalline cohomology and de Rham descent. The vast majority of existing work on infinitesimals and infinitesimal neighbourhoods comes from algebraic geometry. Local diffeomorphism, formally Ã©tale morphismĮmbedding of smooth manifolds into formal duals of R-algebrasĭerivations of smooth functions are vector fields Pullback of differential forms, invariant differential form, Maurer-Cartan form, horizontal differential form, Vector field, multivector field, tangent Lie algebroid ĭifferential forms, de Rham complex, Dolbeault complex Smooth manifold, smooth structure, exotic smooth structureįormal smooth manifold, derived smooth manifold

Infinitesimal space, infinitesimally thickened point, amazing right adjointĭifferentiable manifold, coordinate chart, atlas Geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry From point-set topology to differentiable manifolds

Arithmetic of infinitesimals