Purgatory is about how a crooked will is set straight. It is the cantica in which the souls are no longer what they did, they are ceasing to be it, and the question that runs through it is what a person learns to want. Canto I defines the kingdom by two actions of the spirit: purging oneself and becoming worthy to rise to heaven.
The thesis that organizes everything comes in canto XVII, in Virgil's mouth: neither creator nor creature was ever without love, and what varies is the object and the intensity. Vice, by this reckoning, is disordered love, and for that reason it can be corrected. Hence the seven ledges, which cleanse love of harm to a neighbor (pride, envy and wrath), lukewarm love of the good (sloth), and excessive love of lesser goods (avarice, gluttony and lust).
The cantica's themes unfold from this root. Forgiveness appears in Manfred, in Buonconte's small tear and in the prayer that shortens the wait; fate and free will in Marco Lombardo and in Dante's objection to the doctrine of love; art in Casella, Oderisi, Bonagiunta and Guinizelli, poets who speak of their own craft; politics in the invective against Italy and in the two suns of Rome. At the end, Virgil gives Dante a will "free, upright and sound", and Beatrice holds him to what he has made of it.
The cantica does not end on a thesis. It ends on a climb it does not show, with Dante washed clean but still to climb.