Successfully complete the indicated task to unlock the corresponding Colloseum:

Castle Colloseum: Successfully complete the Sunset Colloseum.
Crystal Colloseum: Successfully complete Waterfall Colloseum.
Magma Colloseum: Successfully complete Sunny Park Colloseum.
Neon Colloseum: Successfully complete Waterfall Colloseum.
Pikachu Glove: Successfully complete Neon Colloseum.
Rank 2 Gateway Colloseum: Successfully complete Gateway Colloseum.
Star View Colloseum: Successfully complete the Castle Colloseum.
Sunny Park Colloseum: Successfully complete Neon Colloseum.