space of sections (changes) in nLab

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Contents

Idea

Given a bundle E→ΣE \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≔∏ ΣE∈H \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 /Σ⟵Σ *⟶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 U∈HU \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

U⟶Γ Σ(E) U \longrightarrow \Gamma_\Sigma(E)

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

Σ×U=Σ *U⟶E. \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

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)