On the duality between homological quantum codes of a hypermap and its dual hypermap

From a given topological hypermap $H$, we define two related hypermaps $H^\triangle$ and $H^\nabla$ as complements of the ordinary dual hypermap $H^*$ along with the concepts of their edge hypermap quantum codes $\mathcal{C}^\triangle$ and $\mathcal{C}^\nabla$. We then show that, when the sets of special darts are naturally corresponded, the duality between the ordinary hypermap quantum code $\mathcal{C}$ from $H$ and the one $\mathcal{C}^*$ from $H^*$ can be greatly simplified to the duality between $\mathcal{C}^\triangle$ and $\mathcal{C}^\nabla$.