nLab
smooth natural numbers

**differential geometry** **synthetic differential geometry** ## Axiomatics ## * smooth topos * infinitesimal space * amazing right adjoint * Kock-Lawvere axiom * integration axiom * microlinear space * synthetic differential supergeometry * super smooth topos * . infinitesimal cohesion * de Rham space * formally smooth morphism, formally unramified morphism, formally étale morphism * jet bundle ## Models ## * Models for Smooth Infinitesimal Analysis * Fermat theory * smooth algebra ($C^\infty$-ring) * smooth locus * smooth manifold, formal smooth manifold, derived smooth manifold * smooth space, diffeological space, Frölicher space * smooth natural numbers * Cahiers topos * smooth ∞-groupoid * synthetic differential ∞-groupoid ## Concepts * tangent bundle, * vector field, tangent Lie algebroid; * differentiation, chain rule * differential forms * differential equation, variational calculus * Euler-Lagrange equation, de Donder-Weyl formalism, variational bicomplex, phase space * connection on a bundle, connection on an ∞-bundle * Riemannian manifold * isometry, Killing vector field ## Theorems * Hadamard lemma * Borel's theorem * Boman's theorem * Whitney extension theorem * Steenrod-Wockel approximation theorem * Poincare lemma * Stokes theorem * de Rham theorem * Chern-Weil theory ## Applications * Lie theory, ∞-Lie theory * Chern-Weil theory, ∞-Chern-Weil theory * gauge theory * ∞-Chern-Simons theory * Klein geometry, Klein 2-geometry, higher Klein geometry * Euclidean geometry, Cartan geometry, higher Cartan geometry * Riemannian geometry * gravity, supergravity

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Contents

Idea

In every smooth topos there is a notion of infinitesimal object and of infinitesimal number. The most common such infinitesimal numbers are nilpotent, but in some special smooth toposes, there is in addition a notion of invertible infinitesimal. In these toposes there is likewise an object of smooth natural numbers, which contains infinite or nonstandard natural numbers? (and whose inverses are invertible infinitesimals).

Some examples of such smooth toposes are discussed at Models for Smooth Infinitesimal Analysis.

The central mechanism

The phenomenon of “smooth” nonstandard natural numbers in a Grothendieck topos arises from the following simple general principle:

Consider any sheaf topos 𝒯=Sh(C)\mathcal{T} = Sh(C) such that

  1. the category Set embeds full and faithfully into 𝒯\mathcal{T}.

    (For smooth toposes we have, if they are “well adapted”, even a full and faithful inclusion of Diff and then the one of Set is the one induced by the inclusion SetDiffSet \hookrightarrow Diff.)

  2. the Grothendieck topology on CC is on each object given by finite covering families.

In such a case there are two objects in 𝒯\mathcal{T} that both look like they should qualify as the internal object of natural number, but that are different:

  1. The image NN of the set \mathbb{N} under the given full and faithful embedding.

    This yields, trivially, a sheaf such that morphisms from any other set into it are given by arbitrary \mathbb{N}-valued functions on this set.

  2. The abstractly defined natural numbers object Δ()\Delta(\mathbb{N}):

    this is the sheafification of the presheaf that is constant on the set \mathbb{N}. A morphism into this presheaf is a constant \mathbb{N}-valued function. And since we are sheafifying, by assumption, with respect to finite covers, a morphism from a set into its sheafification is a function into \mathbb{N} that is constant on each patch of a finite cover of that set and hence is a bounded \mathbb{N}-valued function.

The unbounded functions thus represent infinite? or non-standard? “smooth natural numbers.” In particular, a generalized element nΔ()n \in \Delta(\mathbb{N}) with domain of definition \mathbb{N} (regarded as an object of 𝒯\mathcal{T}) is a bounded sequence of integers, whereas a similarly defined generalized element νN\nu \in N is a possibly unbounded sequence of integers. This is intuitively similar to the unbounded sequences of numbers that represent infinitely large numbers in the ultrafilter approach to nonstandard analysis (a different way of making infinitesimal numbers precise).

The generic nonstandard number

The generic non-standard natural number is the generalized element of NN on the domain of definition C ()/NullTail\ell C^\infty(\mathbb{N})/{NullTail} given by the canonical injection C ()/NullTailN\ell C^\infty(\mathbb{N})/NullTail \to N that is dual to the canonical projection of the ring onto its quotient. Here NullTailNullTail is the ideal of sequences of real numbers that vanish above some integer.

The ring C ()/NullTailC^\infty(\mathbb{N})/NullTail here is a quotient ring of sequences as above, where two sequences are identified if they agree above some integer. So C ()/NullTail\ell C^\infty(\mathbb{N})/NullTail is the smooth locus whose function algebra is similar to a nonstandard extension of \mathbb{R}.

The generic non-standard natural number is discussed on page 252 of the Moerdijk-Reyes book below.

References

See in

chapter VI – there section 1.6 section 2 – and chapter VII.

Revised on January 17, 2010 19:12:01 by Toby Bartels (173.60.119.197)