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Models for Smooth Infinitesimal Analysis

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Synthetic differential geometry

differential geometry

synthetic differential geometry

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This entry is about the book

about models of smooth toposes for synthetic differential geometry that have a full and faithful embedding of the category Diff of smooth manifolds.

Contents

Summary

The book discusses the construction and the properties of smooth toposes (𝒯,R)(\mathcal{T},R) that model the axioms of synthetic differential geometry and are well-adapted to differential geometry in that there is a full and faithful functor Diff𝒯Diff \to \mathcal{T} embedding the category Diff of smooth manifolds into the more general category 𝒯\mathcal{T}.

All models are obtained as categories of sheaves on sites whose underlying category is a subcategory of that of smooth loci.

Models

The following tabulates various models for smooth toposes and lists their properties.

the topos 𝒢\mathcal{G}

the topos \mathcal{F}

the topos 𝒵\mathcal{Z}

The smooth topos 𝒵\mathcal{Z} is that of sheaves on the category 𝕃\mathbb{L} of smooth loci with respect to the Grothendieck topology given by finite open covers of smooth loci.

Definition

𝒵:=Sh finopen(𝕃)\mathcal{Z} := Sh_{fin-open}(\mathbb{L}) is the category of sheaves on the entire site 𝕃\mathbb{L} of smooth loci where the covering sieves of any smooth locus A\ell A are those generated by covering families

{f i:A iA} \{f_i : \ell A_i \to \ell A\}

given by a finite collection of elements (a iA) i=1 n(a_i \in A)_{i=1}^n such that the ideal generated by these elements contains the unit, 1(a 1,,a n)1 \in (a_1, \cdots, a_n), and for each ii a commutative diagram

A i (A{a i 1}) f i A, \array{ \ell A_i &&\stackrel{\simeq}{\to}&& \ell (A\{a_i^{-1}\}) \\ & {}_{\mathllap{f_i}}\searrow && \swarrow \\ && \ell A } \,,

where the right diagonal morphism is the canonical inclusion of a smooth locus corresponding to a smooth ring with one element inverted.

This is in chapter VI, 1. Inversion of elements is described around proposition 1.6 in chapter I.

For instance for A=R=C ()\ell A = R = \ell C^\infty(\mathbb{R}) the real line, a covering family is given by maps from two further copies of the real line f 1,2:RRf_{1,2} : R \to R determined by any two smooth functions a 1,a 1C ()a_1, a_1 \in C^\infty(\mathbb{R}) with support (,1)(-\infty,1) and (1,)(-1,\infty). By proposition 1.6 in chapter I we have C (){a 1 1}=C ((,1))C^\infty(\mathbb{R})\{a_1^{-1}\} = C^\infty((-\infty,1)) and C (){a 2 1}=C ((1,))C^\infty(\mathbb{R})\{a_2^{-1}\} = C^\infty((-1,\infty)). As both these open intervals are diffeomorphic to the real line, and as Diff embeds fully, we have isomorphisms R(C (/a i 1)R \stackrel{\simeq}{\to} \ell(C^\infty(\mathbb{R}/{a_i^{-1}}). Hence the cover defined by (a i,a 2)(a_i, a_2) is the ordinary open cover of the real line by the two open subsets (,1)(-\infty,1) and (1,)(-1,\infty).

Similarly using I.1.6, one finds the general result:

Lemma

Let C (U)/I\ell C^\infty(U)/I be a smooth locus with U nU \subset \mathbb{R}^n open and II an ideal of C (U)C^\infty(U). Then, up to isomorphisms, its covering families are precisely those families

{C (U i)/(IU i)C (U i)/I} i=1 n \{\ell C^\infty(U_i)/(I|U_i) \to \ell C^\infty(U_i)/I\}_{i=1}^n

such that the U iUU_i \subset U are an ordinary open cover of UU.

Equivalently, such families where the (U i)(U_i) need not cover all of UU, but where there is VUV \subset U open such that the (U i)(U_i) together with VV do cover and 1I V1 \in I|_V.

Proof

This is lemma 1.2 in chapter VI.

We now list central properties of this topos.

Proposition

(properties)

For the topos 𝒵\mathcal{Z} the following is true.

  • the Grothendieck topology is subcanonical

    (chapter VI, lemma 1.3)

  • the category Diff of smooth manifolds embeds full and faithfully, Diff𝒵Diff \hookrightarrow \mathcal{Z}

    (chapter VI, corollary 1.4)

  • the general Kock-Lawvere axiom holds

    (chapter VI, 1.9)

  • the integration axiom holds

    (chapter VI, 1.10)

  • it models nonstandard analysis in that

    • since the topology is subcanonical, in particular the smooth locus 𝕀:=(C (0)/(fgerm 0(f)=0))\mathbb{I} := \ell(C^\infty(\mathbb{R}-{0})/(f|germ_0(f) = 0)) of the ring of restrictions of germs of functions at 0 to 0\mathbb{R}-{0} is an object: the object of invertible infinitesimals.

      However, even though this object exists, in the intuitionistic internal logic of the topos one cannot prove that there are any infinitesimal elements : all one can prove is that it is false that there are no elements: ¬¬x:x𝕀\not \not \exists x : x \in \mathbb{I}.

      (chapter VI, section 1.8)

      This changes when one refines to the topos \mathcal{B}, discussed below (section VI.5).

    • due the conditions that covers are finite, the smooth locus N:=C ()N := \ell C^\infty(\mathbb{N}) – which is such that functions to it are arbitrary locally constant \mathbb{N}-valued functions – does not coincide with the natural numbers object of the topos, which is the sheafification of the presheaf constant on Set\mathbb{N} \in Set:

      since covering families are by finite covers it follows that maps into the sheafification of the presheaf constant on \mathbb{N} are bounded smooth \mathbb{N}-valued functions, instead of all such functions.

      (chapter VI, 1.6)

      The object N=C ()N = \ell C^\infty(\mathbb{N}) is called the object of smooth natural numbers . It may be thought of as containing “infinite natural numbers”.

the topos \mathcal{B}

The smooth topos \mathcal{B} may be motivated as a slight refinement of the topos 𝒵\mathcal{Z} designed such that in the internal logic of \mathcal{B} it does become true that for 𝕀\mathbb{I} the object of invertible infinitesimals, we have x:x𝕀\exists x : x \in \mathbb{I}, internally.

Definition

(chapter VI, 5.1)

:=Sh finopen/proj(𝕃)\mathcal{B} := Sh_{fin-open/proj}(\mathbb{L}) is the category of sheaves on the site 𝕃\mathbb{L} of smooth loci with covering sieves given by

  • finite open covers as above for \mathbb{Z}

  • and in addition letting projections A×BA\ell A \times \ell B \to \ell A out of products in 𝕃\mathbb{L} (for B0B \neq 0 if A0A \neq 0) be singleton covers.

(This is not a Grothendieck topology, as it is not closed under composition, but still a coverage.)

Proposition

(properties)

The topos inherits most of the properties of 𝒵\mathcal{Z}, notably:

A main difference is that in \mathcal{B} every smooth locus, i.e. every representable, is an inhabited object. In particular therefore there exist, in the internal logic, elements of the object of invertible infinitesimals:

x𝕀. \exists x \in \mathbb{I} \,.

(chpater VI, prop 5.4).

category: reference

Revised on January 25, 2012 15:22:48 by Zoran Škoda (161.53.130.104)