nLab space of sections

Idea

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

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

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

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

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

$$\Gamma_\Sigma E \coloneqq \prod_{\Sigma} E \;\in\; \mathbf{H}$$

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

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

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

$$U \longrightarrow \Gamma_\Sigma(E)$$

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

$$\Sigma \times U = \Sigma^\ast U \longrightarrow E \,.$$

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

$$\array{ && E \\ & {}^{\mathllap{\phi_U}}\nearrow & \downarrow^{\mathrlap{p}} \\ U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma } \,,$$

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

Properties

Topological vector space structure

Definition

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

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

$$\array{ \Gamma_\Sigma(E) &\overset{p_K^N}{\longrightarrow}& [0,\infty) \\ \Phi &\mapsto& \underset{n \leq N}{max} \left( \underset{x \in K}{sup} {\vert \nabla^n \Phi(x)\vert}\right) \,, }$$

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

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

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

$$\Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E)$$

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

(Bär 14, 2.1, 2.2)

Extension of sections

See at Whitney extension theorem (Roberts-Schmediung 18).

Examples

• The space of section of an universal associated infinity-bundle is the space of homotopy coinvariants of the corresponding infinity-action (see there for more).

Related concepts

• dependent product

• twisted cohomology

• space of states (in geometric quantization)

References

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

• Romeo Brunetti, Klaus Fredenhagen, Pedro Ribeiro, around remark 2.2.1 in Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields (arXiv:1209.2148, spire)

• Christian Bär, Green-hyperbolic operators on globally hyperbolic spacetimes, Communications in Mathematical Physics 333, 1585-1615 (2014) (doi, arXiv:1310.0738)