All you need are your monsters and Colosseum Cards. The number of Cards needed to access battles will vary depending on the difficulty you choose.

You also need to be
level 40 to unlock Faction Colosseums.

Like the Stamina you use in Dungeons and the Adventure Map,
Colosseum Cards refill on their own over time. However, if you want extra Colosseum Cards, you can buy them at the Era Shop!