TITLE: A dual theory to the theory of currents?
QUESTION [6 upvotes]: The k-currents are defined as dual space to the spaces of all smooth k-forms.
(These monsters are used to work with the minimal k-surfaces.)
Assume I want to look at the generalized k-forms;
they can be defined as  certain functionals 
on the space of all smooth k-surfaces with boundary.
Did anybody consider such "generalized k-forms"?
In fact I am looking for such generalized 2-forms on $\mathbb R^n$ with values $so(n)$,
this should be some kind of generalized curvature.
(Instead of integral I should consider parallel translation around the boundary. 
If $n=2$, it gives me a sign-measure.)
But I am also interested in the linear case.

REPLY [2 votes]: There is another notion of generalized form called charges. You can find its definition here :
http://www.math.jussieu.fr/~depauw/preprints/dep-moo-pfe.pdf (they are functionals on the space normal currents with a distribution-like topology). 
Cochains can be represented as $L^1$ forms with $L^1$ exterior derivative, whereas charges are represented as $\omega + d \eta$, where $\omega$ and $\eta$ are continuous forms. Maybe this kind of generalized form is best suited for your purposes.