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.
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)