Let $M\subset\mathbb R^N$ be a compact oriented $n$-(sub)manifold with corners and $\omega$ be an $(n-1)$-form on it. The usual statement of Stokes theorem
$$\int_M d\omega=\int_{\partial M}\omega$$
is proven using the fact that the corners have null measure in $\partial M$ and hence do not contribute to the integral. This justifies the definition of $\partial M$ as just the dimension $n-1$ components of the topological version of $\partial M$ which is the disjoint union of manifolds without boundary of dimension $0,1,\dots ,n-1$.
Let $\mathcal D^p(M)$ be the space of $p$-forms on $M$ with the topology induced by the norm $\lVert\omega\rVert=\sum_I \lVert f_I\rVert_\infty$ where $\omega=\sum_{I}f_I dx_I$. We can define a current of degree $p$ as a continuous linear form
\begin{align*}
t:\mathcal D^{n-p}&\to\mathbb C\\
\alpha&\mapsto T(\alpha)=\left<T, \alpha\right>=\int_M T\wedge\alpha.
\end{align*}
And the differential by the tautological-looking yet necessary $\left< dT,\alpha\right>=(-1)^{p+1}\left<T, d\alpha\right>$. For $p=n$ we likewise define $\int_M T=\int_M T\wedge 1=\left<T, 1\right>$ where $1$ is the $0$-form identified to the smooth function which is constant and equal to $1$. Now I want to have Stokes theorem for forms on $M$, that is: Given an $(n-1)$-current $T$ on $M$ we have
$$\int_M dT=\int_{\partial M}T$$
for an appropriate definition of $\partial M$. This time we can't ignore the corners like we did for integration of smooth forms because null-measure sets do contribute to the integral. It seems nontrivial to figure out what's the correct orientation for the components of $\partial M$ with dimension less than $n-1$. What is the correct definition of $\partial M$, especially the orientation of each component? I'd love to have an answer or a reference to more information about currents and manifolds with corners.
(I think all of this would be also possible without the assumption that $M$ is compact and a submanifold of euclidean space but the statements might be longer.)
Best Answer
This question has been largely answered by Huber 2023:
Proposition 6.3: Let $X$ be a definable $C^p$-manifold with corners for $2\leq p<\infty$. Let $G\subset X$ be a pseudo-oriented compact definable subset of dimension $d$ and $\omega$ a $(d-1)$-form of class $C^1$ in a neighborhood of $G$. Then there's a formal $\mathbb Z$-linear combination $\Sigma$ of pseudo-oriented compact definable subsets of $G$ of dimension $d-1$ such that $$\int_G d\omega=\int_{\Sigma}\omega.$$
See the paper for all the definitions. I haven't completely understood it to be honest, but I know she does answer the crux of my question which was how to orient the boundary of a manifold with corners. The answer being to do a triangulation procedure and orient it as a manifold with corners of one dimension lower. The paper cites a paper by Antoine Julia and his PhD thesis which do something for currents with singularities which is also worthwhile to read.