In the nLab page on [transgression of differential forms](https://ncatlab.org/nlab/show/transgression+of+differential+forms) at definition 2.7 they have



> 
> Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a field bundle over a spacetime $\Sigma$ (def. 2.5), with induced jet bundle $J\_{\Sigma}^{\infty}(E)$.
> For $\Sigma\_r \hookrightarrow \Sigma$ be a submanifold of spacetime of dimension $r \in \mathbb{N}$, then *transgression of variational differential forms to $\Sigma\_r$* is the function
> $$
> \tau\_{\Sigma\_r}: \Omega\_{\Sigma, \mathrm{cp}}^{r, \bullet}(E) \longrightarrow \Omega^{\bullet}\left(\Gamma\_{\Sigma\_r}(E)\right)
> $$
> which sends a differential form $A \in \Omega\_{\Sigma, \text { cp }}^{r, \bullet}(E)$ to the differential form $\tau\_{\Sigma\_r} \in \Omega^{\bullet}\left(\Gamma\_{\Sigma\_r}(E)\right)$ which to a smooth family on field configurations
> $$
> \Phi\_{(-)}: U \times N\_{\Sigma} \Sigma\_r \longrightarrow E
> $$
> assigns the differential form given by first forming the pullback of differential forms along the family of jet prolongation $j\_{\Sigma}^{\infty}\left(\Phi\_{(-)}\right)$followed by the integration of differential forms over $\Sigma\_r$ :
> $$
> \tau\_{\Sigma}A\_{\Phi\_{(-)}}:=\int\_{\Sigma\_r}\left(j\_{\Sigma}^{\infty}\left(\Phi\_{(-)}\right)\right)^\*A \in \Omega^{\bullet}(U) .
> $$
> 
> 
> 


Why is the last expression a differential form? After integration we have a number so isn't it a function?



Why is the transgression of differential forms a form?

> 
> After integration we have a number so isn't it a function?
> 
> 
> 


Differential $k$-forms can be integrated along a submersion with $d$-dimensional fibers, which yields a differential $(k-d)$-form.


Fiberwise integration (alias pushforward) of differential forms is a standard operation, described in many expository texts. A fairly detailed presentation is given in Chapter VII of


* Werner Greub, Stephen Halperin, Ray Vanstone. Connections, Curvature, and Cohomology. Volume I. [De Rham Cohomology of Manifolds and Vector Bundles](https://doi.org/10.1016/S0079-8169(08)62844-5). Pure and Applied Mathematics 47A (1972), Academic Press.


