Duking's house is in the southwestern portion of Pyrite. Duking is the closest thing Pyrite has to a leader or mayor; however, that appears to be because of his size rather than a formal election. Underneath Duking's house was one of the Kids Grid nodes.
Pyrite Colosseum is located in the northernmost part of town; in fact, it is the only building located north of the chasm. Although it appears to be dilapidated, it is still functional. Miror B. once dispensed Shadow Pokémon to Trainers who won challenges in the Colosseum.