In what way is Murder in the Cathedral a tragedy of ambition?
While Becket has honorable qualities and espouses his ambitions as being toward holiness only, his world view permitted competition against the ambitions of the King of England to be an element of sacrificial holiness. In this way is Murder in the Cathedral a tragedy of ambition.
While Becket was in France, he conspired with the King of France and with the Pope of the Roman Catholic Church to find ways to unseat the young King Henry. One avenue of attack proscribed by the Pope was to excommunicate the cardinals who had crowned Henry King. If the cardinals were unworthy, and now defrocked, ministers of God, then the rights they administered, including the coronation of the Prince as King, might be viewed as retroactively (meaning all past and present acts) invalid as violations of God's ordinances: They had no authority, either presently or retroactively, to perform rites in the name of God.
The struggle for power between Becket and Henry is well known to the Cathedral priests and to the local townspeople, especially the women who break their protective anonymity to go to the Cathedral to witness what they hope will be Becket's return to France. The herald holds out hope of a compromised peace to the women and priests without knowing that King Henry had sworn that he would never see Becket alive again, this being a sure indication of what Henry wills the resolution of their power struggle to be. That there was a known power struggle of ambition between Henry and Becket is illustrated by the Tempter's words to Becket:
For us, Church favour would be an advantage,
Blessing of Pope powerful protection
In the fight for liberty. You, my Lord,
In being with us, would fight a good stroke
At once, for England and for Rome,
Ending the tyrannous jurisdiction
Of king's court over bishop's court,
Of king's court over baron's court.
Another critical part of the historical power struggle between Henry and Becket was that Henry appointed Archbishop of Canterbury under the firm belief that Becket would obey Henry's every wish and command and that the Church of England would therefore not be an autonomous power able to challenge him but would be a docile collaborator in all of Henry's plans, schemes and mechanization.