Possibly, but there is no solid historical evidence for it. Christians were certainly put to death in Roman arenas, and ancient writers describe it. What cannot be shown is that any particular martyr died in the Colosseum itself. The Catholic Encyclopedia, following the Bollandist scholar Hippolyte Delehaye, concludes that we know nothing on the subject one way or the other, and that medieval Rome had no tradition linking the martyrs with the building. Catholics honor the Colosseum as a symbol of all the martyrs, not as a proven site of martyrdom.
The association is a devotional one that grew in later centuries. Benedict XIV erected Stations of the Cross in the arena in the eighteenth century, at the urging of St. Leonard of Port Maurice, and the papal Way of the Cross is customarily held there on Good Friday. St. John Paul II chose it for the ecumenical commemoration of the witnesses to the faith of the twentieth century in May 2000.
The archaeologist Fabrizio Bisconti said in a 2013 interview that no literary or other source gives certainty of martyrdoms in the Colosseum. Even the well-known story that St. Ignatius of Antioch died there rests on evidence the Catholic Encyclopedia calls far from decisive. The same caution applies to many of the written legends of the Roman martyrs, much of which the Catholic Encyclopedia ranks as historical romance built around a few real facts; the burial places and the early veneration are usually on firmer ground.
Good to know before you go: pilgrims who want places where Roman martyrs are known to be buried should go to the catacombs and to the basilicas of St. Peter and St. Paul. From the Colosseum, three churches tied to the martyrs stand within a short walk on and around the Caelian Hill: San Clemente, Santi Quattro Coronati and Santo Stefano Rotondo.
Related questions, answered
Why is there a cross in the Colosseum?
It marks the arena as a place of Christian prayer and a memorial of the martyrs. Benedict XIV erected Stations of the Cross there in the eighteenth century (they were removed in 1874), and the papal Way of the Cross is customarily held at the Colosseum on Good Friday. The cross is a memorial; it is not a claim about a specific martyrdom.
Source: Catholic Encyclopedia: The Coliseum
Where in Rome were Christians martyred, then?
The best-attested case is the persecution under Nero, in which, by ancient tradition, St. Peter died in Nero's circus on the Vatican Hill, beside the present basilica; St. Paul was beheaded outside the city, by tradition at Tre Fontane. For most Roman martyrs the place of burial is better known than the place of death: St. Callixtus on the Appian Way for the popes and martyrs of the third century, St. Lawrence Outside the Walls for the deacon martyred in 258, St. Praxedes, and the two churches of St. Agnes, in Piazza Navona on the traditional site of her death and on the Via Nomentana where, by tradition, she is buried.
Source: Opera Romana Pellegrinaggi: What to see on a pilgrimage to RomeCatacombs of St. Callixtus (official site)Catholic Encyclopedia: St. Lawrence
