Penitence
Much of the Purgatorio is concerned with the theme of penitence; the name Purgatory itself comes from the term for cleansing ("purgation"). Dante works to develop a complex representation of the concept that goes beyond (while still including) the concept of mere punishment. Most ostensibly, each terrace features a contrapasso punishment associated with one of the seven capital sins. The envious, for example, have their eyes sewn shut in an ironic reversal to punish their envious eyes during life; yet the punishment is also meant to be a sort of corrective measure, as now they can only see the sun, symbolizing God, through their eyes. Penitence, while connected to the motif of the repayment of debts, also becomes a way for the will to be corrected.
Similarly, Dante's journey through Purgatory reflects his own journey towards penitence. A sensitive reading reveals that as Dante moves from terrace to terrace, he seems to attend to the sinners and change his behavior slightly as his ascends. Indeed, this change is symbolized by the "P"s slowly being removed from his forehead. And finally, the climax of the Purgatorio, his interaction with Beatrice, is in a significant way a dramatization of penitence, in which Dante's tears pay the debt of his punishment.
Love
Fueling both sin and virtue, love takes a central place in the Purgatorio. This becomes most evident in Virgil's speech in canto XVII, in which he explains that there are two types of love, one natural and one mental. All forms of sin come from mental love, which is created by humans and can err. Similarly, all forms of virtue come from natural love, which is divinely inspired. Ultimately, this definition of sin and virtue, while having precedents in medieval thinkers like St. Thomas Aquinas, is somewhat particular to Dante and reveals the extent to which his Commedia is also a work of philosophy and theology, attempting to understand the nature of the divine and the human. Indeed, we might even be surprised to see the extent to which Dante puts a concept so close to desire in the center of human action, and yet, Dante's focus on love does exactly that. Even his "sweet new style," explored in canto XXIV, is said to follow "whatever Love may dictate." Even Dante's poetry is led by love.
History and Politics
On a broad scale, Dante's involvement in history and politics reveals itself on each terrace of Purgatory. Most obviously, he draws on figures from history to depict examples both of the sinful and the virtuous, framing each sin in both biblical and classical history. More importantly, Dante's Commedia comments on the history and politics of his own time. Ravaged by division and civil war, medieval Italy was a turbulent collection of cities and factions, and this chaos left its mark on Dante. He himself was a political exile, forced out of his native Florence by the Black Guelfs. Not only do Dante's politics appear in who he presents in Purgatory, but they also appear in how he reacts to them. Indeed, moving beyond the Guelf/Ghibelline or White Guelf/Black Guelf binaries which defined much of Italian politics, Dante reveals his own political thought, which seems to have been critical of most factions while still in support of using the Holy Roman Empire to moderate the power and corruption of the Church. This political bent is most prominent in the final cantos of the Purgatorio, where Dante draws on the political and visionary poetry of Isaiah and the prophetic books of the Bible to allegorize recent history while still suggesting his hope for future peace.
Guidance
Just as navigation is a persistent motif throughout the Commedia, guidance appears as an overarching theme that defines much of the poem. This theme works on both a literal level and a metaphorical level throughout the narrative; looking to Virgil alone, we see how he physically guides Dante through the space of Purgatory while also offering spiritual guidance. This doubled guidance, while working as a sort of extended allegory, is importantly tied to a Christian focus on pilgrimage in the middle ages; Dante takes a real social practice in which Christians took physical and spiritual journeys and transports it into an imaginative and poetic context.
Beyond his own personal journey and need for guidance, Dante reveals his belief in a need for guidance on a larger scale, revealing how his poem, seemingly self-centered, might have a broader social purpose. In canto XVI, Marco the Lombard suggests that the current cause of social and political disorder is "failed guidance." Dante's choice of "guidance" rather than "governance" or politics gestures to the fact that the type of guidance he needs in the Commedia may be useful and indeed needed on a broad scale.
Women and Virtue
Although it would be anachronistic (and in many ways reductive) to consider Dante any sort of feminist, attention should be pay to the prominent presence of women in the Commedia. In the Purgatorio, we should attend to the use of Mary as the first example of whatever virtue is in opposition to the sin being punished on each terrace. Her images and actions, by dint of repetition, give structure to the Purgatorio and present her as the consummate figure of virtue. This reflects at once Mary's importance in the Catholic church and Dante's own apparent focus on mothers as moral guides; Statius, for example, refers to the Aeneid as his "mamma," revealing a surprising elision of the importance of father figures.
Of similar importance is the women of Dante's dreams. Beginning with Saint Lucy's slightly encoded presence in the first dream, women take center stage; in the first, the eagle (typically a male symbol) represents Saint Lucy. In the second, a woman is represented as a seductive siren, beckoning Dante to sin. Importantly, Dante is rescued by Virgil and a "holy woman," but we should be aware of Dante's use of misogynist tropes. In the third, Leah and Rachel represent two different kinds of virtuous life. Leading into the confrontation with Beatrice, these dreams paint a complex picture of women as the shaping and guiding forces in much of Dante's life, whether for good or ill.
Philosophy and Theology
Although today we consider philosophy, theology, and poetry as distinct forms of writing and thinking, many medieval writers divided these genres much less rigidly. Works like The Consolation of Philosophy by Boethius mix poetry, narrative, and philosophical analysis in a single work, and Dante's Commedia is no different. The Purgatorio tackles a series of difficult philosophical and theological questions, but perhaps most conspicuously it attempts to untangle the nature of love and free will. Ultimately, Virgil's monologue, explained above, develops a concept of love and presents free will as primarily negative, meaning humans are able to choose to abstain from action. Another important thing to consider is that philosophy and science were less rigidly divided. As such, Dante explores questions such as the development of the fetus as it relates to the human soul. These considerations, in sum, allow the Purgatorio to reveal the complex intersection of different kinds of knowledge, undoing some of the divisions we hold today and showing how imaginative poetry is able to link them together.
Hope
One of the most recognizable images from the Inferno is the inscription above the entrance to Hell: "ABANDON EVERY HOPE, WHO ENTER HERE." If Hell is defined by an absence of hope, the Purgatorio is defined by the persistent presence of hope. In the third canto, "eternal love" is assured by hope as long as there is "a thread of green." This image of hope then returns as the Garden of Eden is rejuvenated by the Griffin and Beatrice. Even further, the consistent presence of the sun across the Purgatorio, in contrast to the glum darkness of the Inferno, is a reminder of God's presence, and it offers a sort of hope as Virgil and Dante ascend. Indeed, even the penitents' responses to their punishment reflect an atmosphere of hope. Despite their suffering, some refer to their punishment as a "solace," suggesting the overwhelming hopeful attitude of the saved souls.