If the boundary is adjoint to the differential, what is the "coboundary" adjoint to the codifferential in the continuum?

Question

For a smooth manifold, Stoke's theorem says that the differential/exterior derivative $\mathrm{d}$ is adjoint to the boundary operator $\partial$, i.e. $$\int_{\partial U} \omega = \int_{U} \mathrm{d}\omega.$$ where $\omega$ is a differential form. Given an inner product of forms (e.g. induced by a metric) $\langle\cdot,\cdot\rangle$, the codifferential $\delta$ can be defined as the adjoint of $\mathrm{d}$ with respect to the inner product, $$\langle \mathrm{d}\alpha,\beta \rangle = \langle \alpha, \delta \beta\rangle.$$ My question is, is there a coboundary $\partial^\dagger$ such that we could write a dual version of Stoke's theorem, $$\int_{\partial^\dagger U}\omega = \int_U \delta \omega.$$ I am wondering if there is a meaningful version of this in the continuum. Naively it seems like the answer is ``no'', but it feels like $\partial^\dagger$ ought to have some sort of meaning. One could certainly define such a thing for chains, say on a simplicial complex.