mapping space

# Contents

## Idea

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.

## 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.

## References

Discussion of spaces of smooth sections is in

Revised on September 15, 2017 03:36:58 by David Corfield (31.185.156.112)