space of sections in nLab

Contents

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

(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).

• 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)