microlinear space in nLab

Context

Synthetic differential geometry

• Idea
• Definition
• Construction and Proof
• Examples
• References

Definition

(microlinear space)

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

Proposition

(fiberwise linearity of tangent bundle)

For every microlinear space XX, the tangent bundle X D→XX^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={ϵ∈R|ϵ 2=0}D = \{\epsilon \in R| \epsilon^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 satisfied in the smooth topos 𝒯\mathcal{T}. Since XX is microlinear, also the canonical map

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

  is an isomorphism. With Id×Id:D→D(2)\Id \times Id : D \to D(2) the diagonal map, we then define the fiberwise addition X D× XX D→X 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 D→r −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 2∈X 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):d↦r −1(v 1,v 2)(d,d)(v_1 + v_2) : d \mapsto r^{-1}(v_1,v_2)(d,d).

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

  (α⋅v):d↦v(α⋅d)(\alpha \cdot v) : d \mapsto v (\alpha \cdot d).

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

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:𝒯→𝒯(-)^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}

Proof

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

For 𝒵\mathcal{Z} and ℬ\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=ℓC ∞(ℝ)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 ℓA\ell A of 𝕃\mathbb{L} satisfies the microlinearity axioms in 𝕃\mathbb{L} in that for each cocone Δ→Δ c:J→𝕃\Delta \to \Delta_c : J \to \mathbb{L} of infinitesimal objects such that R Δ c≃lim j∈JR Δ jR^{\Delta_c} \simeq \lim_{j \in J} R^{\Delta_j} we also have (ℓA) Δ c≃lim j∈J(ℓA) Δ j(\ell A)^{\Delta_c} \simeq \lim_{j \in J} (\ell A)^{\Delta_j}. Now, the Yoneda embedding Y:𝕃→PSh(C)Y : \mathbb{L} \to PSh(C) preserves limits and exponentials. Since the Grothendieck topology in question is subcanonical, Y((ℓA) Δ j)Y((\ell A)^{\Delta_j}) is in Sh(C)Sh(C) and hence is the exponential object Y(ℓA) 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(ℓA) Δ c≃lim j∈JY(ℓA) Y(Δ j)Y(\ell A)^{\Delta_c} \simeq \lim_{j \in J} Y(\ell A)^{Y(\Delta_j)} and hence all Y(ℓA)Y(\ell A) are microlinear in 𝒵\mathcal{Z} and ℬ\mathcal{B}.