[enriquto]

stokes theorem is often written (in a continuous setting) as

    < dM, F > = < M, dF >

(the adjoint of the boundary operator is the exterior derivative). In the discrete case, these integrals are actually products of matrices. There is nothing too deep here.

Edit: regarding the "true by definition" issue, you can do one of two things. (1) Build the definition of the boundary and exterior derivative independently and then verify that Stokes theorem holds. Or (2) build the definition of only one of these two operators, and then define the other one as its adjoint. In that second case, Stokes theorem is true by definition. But it doesn't matter too much, these are just two different ways to write the same thing.