nLab
space of sections

Contents

Idea

Given a bundle EΣE \overset{}{\to} \Sigma, then its space of sections is like a mapping space, but relative to the base space Σ\Sigma.

Formally this is given by the dependent product construction. See at section – In terms of dependent product and at dependent product – In terms of spaces of sections.

Definition

Let H\mathbf{H} be a topos (for instance H=\mathbf{H} =SmoothSet) or (∞,1)-topos (for instance H=\mathbf{H} = Smooth∞Grpd) and consider

[E p Σ]:H /Σ \left[ \,\, \array{ E \\ {}^{\mathllap{p}}\downarrow \\ \Sigma } \right] \;\colon\; \mathbf{H}_{/\Sigma}

a bundle in H\mathbf{H}, regarded as an object in the slice topos/slice (∞,1)-topos.

Then the space of sections Γ Σ(E)\Gamma_\Sigma(E) of this bundle is the dependent product

Γ ΣE ΣEH \Gamma_\Sigma E \coloneqq \prod_{\Sigma} E \;\in\; \mathbf{H}

hence the image of the bundle under the right adjoint Σ *\Sigma_\ast in the base change adjoint triple

H /ΣΣ *Σ *Σ !H,. \mathbf{H}_{/\Sigma} \underoverset {\underset{\Sigma_\ast}{\longrightarrow}} {\overset{\Sigma_!}{\longrightarrow}} {\overset{\Sigma^\ast}{\longleftarrow}} \mathbf{H} ,.

By adjunction this means that for UHU \in \mathbf{H} a test object, then a UU-parameterized family of sections of EE, hence a morphism in H\mathbf{H} of the form

UΓ Σ(E) U \longrightarrow \Gamma_\Sigma(E)

is equivalently a morphism in H /Σ\mathbf{H}_{/\Sigma} of the form

Σ×U=Σ *UE. \Sigma \times U = \Sigma^\ast U \longrightarrow E \,.

This is equivalently a diagram in H\mathbf{H} of the form

E ϕ U p U×Σ pr 2 Σ, \array{ && E \\ & {}^{\mathllap{\phi_U}}\nearrow & \downarrow^{\mathrlap{p}} \\ U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma } \,,

where the right and bottom morphisms are fixed, and where ϕ U\phi_U (and the 2-cell filling the diagram) is, manifestly, the UU-parameterized family of sections.

Properties

Tological vector space structure

Definition

(Fréchet topological vector space on spaces of smooth sections of a smooth vector bundle)

Let EfbΣE \overset{fb}{\to} \Sigma be a smooth vector bundle. On its real vector space Γ Σ(E)\Gamma_\Sigma(E) of smooth sections consider the seminorms indexed by a compact subset KΣK \subset \Sigma and a natural number NN \in \mathbb{N} and given by

Γ Σ(E) p K N [0,) Φ maxnN(supxK| nΦ(x)|), \array{ \Gamma_\Sigma(E) &\overset{p_K^N}{\longrightarrow}& [0,\infty) \\ \Phi &\mapsto& \underset{n \leq N}{max} \left( \underset{x \in K}{sup} {\vert \nabla^n \Phi(x)\vert}\right) \,, }

where on the right we have the absolute values of the covariant derivatives of Φ\Phi for any fixed choice of connection on EE and norm on the tensor product of vector bundles (T *Σ) Σ n ΣE(T^\ast \Sigma)^{\otimes_\Sigma^n} \otimes_\Sigma E .

This makes Γ Σ(E)\Gamma_\Sigma(E) a Fréchet topological vector space.

For KΣK \subset \Sigma any closed subset then the sub-space of sections

Γ Σ,K(E)Γ Σ(E) \Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E)

of sections whose support is inside KK becomes a Fréchet topological vector spaces with the induced subspace topology, which makes these be closed subspaces.

(Bär 14, 2.1, 2.2)

Examples

References

The topological vector space on spaces of smooth sections is discussed in

Revised on November 13, 2017 08:27:05 by Urs Schreiber (94.220.55.151)