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I have created an entry transgression of differential forms that discusses the concept using the topos of smooth sets. Apart from the traditional definition as $\tau_{\Sigma} \coloneqq \int_\Sigma ev^\ast$ the entry considers the formulation as
$\tau_\Sigma = \int_\Sigma [\Sigma,-]$which simply forms the internal hom into the classifying map $X \to \mathbf{\Omega}^n$ of a differential form. I have spelled out the proof that the two definitions are equivalent.
Then the entry contains statement and proof of the situation of “relative” transgression over manifolds with boundary. (This is what yields, when applied to Lepage forms, Lagrangian correspondences between the phase spaces with respect to different Cauchy surfaces, which is what I currently need this material for in the exposition at A first idea of quantum fields.)
Finally there are two examples, a simplistic one and an simple but interesting one related to Chern-Simons theory. These two examples I had kept for a long time already at geometry of physics – integration in the section “Transgression”. That section I have now expanded accordingly, its content now coincides with the entry transgression of differential forms.
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