Short answers, pulled from the story.
Ludus in ancient Rome could refer to a primary school, a gladiator training compound, a board game, or festival games tied to religious observances. All of these meanings fall within the Latin semantic field of play, game, sport, and training.
Children at a Roman ludus studied mathematics, reading, writing, poetry, geometry, and sometimes rhetoric. Boys and girls both attended, with school running from around six o'clock in the morning until just after midday.
The teacher at a Roman ludus was called the ludi magister. That person was often an educated slave or freedman rather than a freeborn Roman citizen.
Gladiator training schools in ancient Rome were called ludi. Two named examples are the Ludus Magnus and the Ludus Dacicus.
Two named board games carried the term ludus: ludus latrunculorum and ludus duodecim scriptorum. Games played with knucklebones, known as astragali, also fell under the same word.
Roman poets used ludus to frame verse writing as a form of play, associating it with the stylistic elegance of the Alexandrian literary tradition and with erotic poetry. Scholar Michele Lowrie notes that poetic play in this sense denoted both Alexandrian-style elegance and erotic verse.