The Coliseum is a location of the Valuan empire, it is reached after Vyse and party venture into the Valua Catacombs, and defeat Bleigock. In-game it appears to be located next to Upper city behind the Grand Palace.
The Coliseum serves as the only form of entertainment for poor Lower city Valuans, as one of them mentions:"it feels good to see people in a worse position than us". This seems to be the same for Upper city Valuans but for them it is just another form of indulgement. Overall it is a way for the Valuan Empire to impose terror on it's enemies as it is later mentioned that the Nasultan and his family were executed there.
One of Aika's weapons, the Valuarang, suggest that the Coliseum is also used for competitions.