nLab
microlinear space

Context

Synthetic differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

In traditional differential geometry a smooth manifold may be thought of as a “locally linear space”: a space that is locally isomorphic to a vector space n\simeq \mathbb{R}^n.

In the broader context of synthetic differential geometry there may exist spaces — in a smooth topos 𝒯\mathcal{T} with line object RR — considerably more general than manifolds. While for all of them there is a notion of tangent bundle TX:=X DT X : = X^D (sometimes called a synthetic tangent bundle, with DD the infinitesimal interval), not all such tangent bundles necessarily have RR-linear fibers!

A microlinear space is essentially an object XX in a smooth topos, such that its tangent bundle does have RR-linear fibers.

In fact the definition is a bit stronger than that, but the main point in practice of microlinearity is that the linearity of the fibers of the tangent bundle allows to apply most of the familiar constructions in differential geometry to these spaces.

Definition

Definition

(microlinear space)

Let 𝒯\mathcal{T} be a smooth topos with line object RR. An object X𝒯X \in \mathcal{T} is a microlinear space if for each diagram Δ:J𝒯\Delta : J \to \mathcal{T} of infinitesimal spaces in 𝒯\mathcal{T} and for each cocone ΔΔ c\Delta \to \Delta_c under it such that homming into RR produces a limit diagram , R c Δlim jJR Δ jR^\Delta_c \simeq \lim_{j \in J} R^{\Delta_j}, also homming into XX produces a limit diagram: X c Δlim jJX Δ jX^\Delta_c \simeq \lim_{j \in J} X^{\Delta_j}.

The main point of this definition is the following property.

Proposition

(fiberwise linearity of tangent bundle)

For every microlinear space XX, the tangent bundle X DXX^D \to X has a natural fiberwise RR-module-structure.

Construction and Proof

We describe first the addition of tangent vectors, then the RR-action on them and then prove that this is a module-structure.

  • Addition With D={ϵR|x 2=0}D = \{\epsilon \in R| x^2 = 0 \} the infinitesimal interval and D(2)={(ϵ 1,ϵ 2)R×R|ϵ i 2=0}D(2) = \{(\epsilon_1, \epsilon_2) \in R \times R | \epsilon_i^2 = 0\} we have a cocone

    D(2) D D * \array{ D(2) &\leftarrow& D \\ \uparrow && \uparrow \\ D &\leftarrow& {*} }

    such that

    R D(2) R D R D R \array{ R^{D(2)} &\to & R^D \\ \downarrow && \downarrow \\ R^D &\to& R }

    is a limit cone by the Kock-Lawvere axiom. Since XX is microlinear, also the canonical map

    r:X D(2)X D× XX D r : X^{D(2)} \to X^D \times_X X^D

    is an isomorphism. With Id×Id:DD(2)\Id \times Id : D \to D(2) the diagonal map, we then define the fiberwise addition X D× XX DX DX^D \times_X X^D \to X^D in the tangent bundle X DX^D to be given by the map

    +:X D× XX Dr 1X D(2)X Id×IdX D. + : X^D \times_X X^D \stackrel{r^{-1}}{\to} X^{D(2)} \stackrel{X^{Id \times Id}}{\to} X^D \,.

    On elements, this sends two elements v 1,v 2X Dv_1, v_2 \in X^D in the same fiber to the element v 1+v 2v_1 + v_2 of X DX^D given by the map (v 1+v 2):dr 1(v 1,v 2)(d,d)(v_1 + v_2) : d \mapsto r^{-1}(v_1,v_2)(d,d).

  • Multiplication :R×X DX D \cdot : R \times X^D \to X^D is defined componentwise by

(αv):dv(αd)(\alpha \cdot v) : d \mapsto v (\alpha \cdot d).

One checks that this is indeed unital, associative and distributive. …

Examples

A large class of examples is implied by the following proposition.

Proposition

(closedness of the collection of microlinear spaces)

In every smooth topos (𝒯,R)(\mathcal{T},R) we have the following.

  1. The standard line RR is microlinear.

  2. The collection of microlinear spaces is closed under limits in 𝒯\mathcal{T}:

    for X=lim iX iX = \lim_i X_i a limit of microlinear spaces X iX_i, also XX is microlinear.

  3. Mapping spaces into microlinear spaces are microlinear: for XX any microlinear space and Σ\Sigma any space, also the internal hom X ΣX^\Sigma is microlinear.

Proof

This is obvious from the standard properties of limits and the fact that the internal hom-functor () Y:𝒯𝒯(-)^Y : \mathcal{T} \to \mathcal{T} preserves limits. (See limits and colimits by example if you don’t find it obvious.)

  1. by definition

  2. Let Δ\Delta be the tip of a cocone Δ j\Delta_j of infinitesimal spaces such that lim jR Δ j=R Δ\lim_j R^{\Delta_j} = R^\Delta. Then

    X Δ =(lim iX i) Δ lim iX i Δ lim ilim jX i Δ j lim jlim iX i Δ j lim jX Δ j \begin{aligned} X^{\Delta} &= (\lim_i X_i)^\Delta \\ &\simeq \lim_i X_i^{\Delta} \\ & \simeq \lim_i \lim_j X_i^{\Delta_j} \\ & \simeq \lim_j \lim_i X_i^{\Delta_j} \\ & \simeq \lim_j X^{\Delta_j} \end{aligned}
  3. with Δ\Delta as above we have (writing [A,B][A,B] for the internal hom otherwise equivalently denoted B AB^A)

    [Δ,[Σ,X]] [Δ×Σ,X] [Σ,[Δ,X]] [Σ,lim j[Δ j,X]] lim j[Σ,[Δ j,X]] lim j[Δ j,[Σ,X]] \begin{aligned} [\Delta, [\Sigma, X]] & \simeq [\Delta \times \Sigma, X] \\ & \simeq [\Sigma, [\Delta, X]] \\ & \simeq [\Sigma, \lim_j [\Delta_j, X]] \\ & \simeq \lim_j [\Sigma, [\Delta_j, X]] \\ & \simeq \lim_j [\Delta_j, [\Sigma, X]] \end{aligned}
Proposition

(microlinear loci)

Let \mathcal{F}, 𝒢,𝒵,\mathcal{G}, \mathcal{Z}, \mathcal{B} be the smooth toposes of the same name that are discussed in detail in MSIA, capter III. These are constructed there as categories of sheaves on a subcategory of the category 𝕃=(C Ring fin)\mathbb{L} = (C^\infty Ring^{fin}) of smooth loci.

All representable objects in these smooth toposes are microlinear.

Proof

For \mathcal{F} and 𝒢\mathcal{G} this is the statement of MSIA, chapter V, section 7.1.

For 𝒵\mathcal{Z} and \mathcal{B} the argument is similarly easy:

These are categories of sheaves on the full category 𝕃=(C Ring fin) op\mathbb{L} = (C^\infty Ring^{fin})^{op}. The line object RR is representable in each case, R=C ()R = \ell C^\infty(\mathbb{R}). Every object in 𝕃\mathbb{L} is a limit (not necessarily finite) over copies of RR in 𝕃\mathbb{L}. Accordingly, every object A\ell A of 𝕃\mathbb{L} satisfies the microlinearity axioms in 𝕃\mathbb{L} in that for each cocone ΔΔ c:J𝕃\Delta \to \Delta_c : J \to \mathbb{L} of infinitesimal objects such that R Δ clim jJR Δ jR^{\Delta_c} \simeq \lim_{j \in J} R^{\Delta_j} we also have (A) Δ clim jJ(A) Δ j(\ell A)^{\Delta_c} \simeq \lim_{j \in J} (\ell A)^{\Delta_j}. Now, the Yoneda embedding Y:𝕃PSh(C)Y : \mathbb{L} \to PSh(C) preserves limits and exponentials. Since the Grothendieck topology in question is subcanonical, Y((A) Δ j)Y((\ell A)^{\Delta_j}) is in Sh(C)Sh(C) and hence is the exponential object Y(A) YΔ jY(\ell A)^{Y \Delta_j} there. Finally, the finite limit over JJ is preserved by the reflection PSh(C)Sh(C)PSh(C) \to Sh(C) (sheafification, which acts trivially on our representables), so Y(A) Δ clim jJY(A) Y(Δ j)Y(\ell A)^{\Delta_c} \simeq \lim_{j \in J} Y(\ell A)^{Y(\Delta_j)} and hence all Y(A)Y(\ell A) are microlinear in 𝒵\mathcal{Z} and \mathcal{B}.

References

The notion of microlinear space in the above fashion is due to

  • F. Bergeron, (1980)

and was studied further under the name strong infinitesimal linearity

  • Anders Kock, R. Lavendhomme, Strong infinitesimal linearity, with applications to string difference and affine connections, Cahiers de Top. 25 (1984)

This is similar to but stronger than the earlier “condition (E)” given in

  • Demazure (1970)

which apparently was also called “infinitesimal linearity” (without the “strong”).

Spaces satisfying this condition were called infinitesimally linear spaces, for instance in

The later re-typing of that book

contains in its appendix D the definition of microlinearity as above.

A comprehensive discussion of microlinearity is in chapter V, section 1 of

Revised on December 27, 2013 04:38:00 by Urs Schreiber (89.204.138.168)