An example of the complications in building exterior derivative operators in the presence of boundaries. Using circumcentric subdivision one can identify ‘dual edges’ with ’90 deg. rotated edges’ and build a ‘boundary operator for dual 2d-cells’ $\partial_2^*$ as the transpose of the primal boundary operator for edges $\partial_1$, with the following relation $\partial_2^*=-\partial_1$. However such a relation cannot be set up for $\partial_1^*$ since one would need to add the midpoints of the boundary edges as vertices, and the sizes of $\partial_1^*$ and $\partial _2^T$ would no longer match.
