In the theory of CW complexes, the incidence coefficient between a cell $\sigma$ of dimension $k$ and a cell $\tau$ of dimension $k - 1$ of a complex is defined as the topological degree of the map $\phi$ that attaches $\tau$ into $\sigma$, or 0 if $\tau$ is not contained in $\sigma$. If we write the incidence number as $[\sigma, \tau]$, we can then define the boundary operator as $$\partial\sigma = \sum_\tau [\sigma, \tau]\tau$$ where the sum is over all $k - 1$-dimensional cells of the complex.
By contrast, in geometric or homological integration theory, the chain modules are spaces of currents. We define boundary operators by how they act on differential forms: $$\langle \partial \sigma, u\rangle = \langle \sigma, du\rangle$$ where $u$ is a $k$-form, by analogy with Stokes' theorem. Can we define an analogous notion of incidence number for currents? I'm fine with an answer that requires restriction to e.g. currents of finite mass with finite boundary mass.
I suspect that this is possible with some kind of Riesz representation trick but I haven't seen it written out anywhere. I feel like you could do it by thinking of some appropriate subspace of currents as elements of the duals of Sobolev spaces of differential forms and then use the Hilbert space structure.