hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
A space of sections is like a mapping space, but relative to a base.
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.
Discussion of spaces of smooth sections is in