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-Delta and H-del as complements of the ordinary dual hypermap H-* along with the concepts of their edge hypermap quantum codes C-Delta and C-del. We then show that, when the sets of special darts are naturally related, the duality between the two ordinary hypermap quantum codes, i.e., C from H and C-* from H-* can be simplified to the duality between C-Delta and C-del.

Automated summary

This article introduces the definitions of the given topological hypermap H through the complementary hypermaps H-Delta and H-del of the ordinary dual hypermap H-*, as well as their edge hypermap quantum codes C-Delta and C-del. Then it proves that when there is a natural relationship between the sets of special darts, the duality between the two ordinary hypermap quantum codes, i.e., C from H and C-* from H-*, can be simplified to the duality between C-Delta and C-del.