hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
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.
Let $\mathbf{H}$ be a topos (for instance $\mathbf{H} =$SmoothSet) or (∞,1)-topos (for instance $\mathbf{H} =$ Smooth∞Grpd) and consider
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
hence the image of the bundle under the right adjoint $\Sigma_\ast$ in the base change adjoint triple
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
is equivalently a morphism in $\mathbf{H}_{/\Sigma}$ of the form
This is equivalently a diagram in $\mathbf{H}$ of the form
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.
(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
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
of sections whose support is inside $K$ becomes a Fréchet topological vector spaces with the induced subspace topology, which makes these be closed subspaces.
See at Whitney extension theorem (Roberts-Schmediung 18).
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)
Last revised on March 4, 2019 at 13:55:31. See the history of this page for a list of all contributions to it.