The Infirmary is where your gladiators are able to receive some much needed medical attention, following a battle. It is best to have your gladiators at full health for their battles in order to increase the chances of victory. The infirmary first becomes available when the HQ is at level 1 after your first fight. By upgrading your infirmary, you increase the number of gladiators who can be treated at once. You also decrease the amount of time it takes to heal a gladiator as well as gaining XP points to increase your city level. In order to upgrade your infirmary, the HQ has to be at a certain level.