The Name of the Rose is set in Piedmont in 1327. That land, south of the Alps, is the gateway between the Languedoc and Italy, an important route for merchants, pilgrims and ideas.

Not far north of that route is Lyons where, in the later 12th century, Valdes, a wealthy merchant, gave up all his worldly possessions to pursue a life of poverty and preaching. To live life as the apostles had: Apostolica Vita. Soon, Valdes had many followers in Lyons, and The Poor Men of Lyons, or the Waldensians, were born.

If we jump forward in time, to the late 13th/early 14th centuries, to the primary heresy discussed in The Name of the Rose, the Dulcinians, we can also locate a founding figure, or rather, founding figures: Gerard Segerelli, followed by Fra Dulcino, and both were inspired by the apocalyptic teachings of Joachim of Fiore a century earlier. We do not know with certainty the "class" backgrounds of Segerelli or Dulcino but tradition states that Segerelli was relatively low born, and Dulcino was relatively high born. Nevertheless, both were inspired by the teaching of St Francis, the son of a wealthy merchant, and eventually their methods of poverty were declared heretical.

Ultimately, both of these groups should be considered as reformers. Both wanted elements of the Church to behave in a way that was more a kin to the teachings of Christ and the apostles in the New Testament. Valdes in the pre-friar era, and the Dulcinians in the post-friar era.

From the perspective of the Church, the followers of these groups were rusticani, or Idiotae et Illiterati, it is possible that this refers to peasants, but it is more likely to mean ignorant of Latin and without formal training. However, it is evident from evidence regarding groups like the "Cathars" that support could be found amongst noblemen, and not just the peasantry. Moreover, as Valdes demonstrates some "heretics" would have been members of the "emerging middle class" and through guilt regarding their new wealth decided to lead a life of poverty. Eventually being considered heretical.

It is important to note that a heresy, becomes a heresy because someone with the authority to label it a heresy labels it as such. These people didn't set out to be heretics, they set out to be better Christians.

It is easy to associate these groups, particularly with regards to their poverty, with the peasantry. However, a key component was the "voluntary" nature of their poverty. They had sacrificed something in becoming poor. St Francis of Assisi, the son of a wealthy merchant, gave up his fortune. There was something noble in that behaviour that people of the time, of all backgrounds, could get behind.

So in answer to your question, I would argue the heretical groups of this time were composed of too many different people, with too many different backgrounds for them to be labelled as "peasant rebellions". Former members of the clergy, noblemen and women, wealthy merchants, artisans, weavers, farmers, and importantly women (some of these groups gave them permission to preach), all played their roles in these movements.

Whilst not peasant rebellions as such, they were theological reformers and, once condemned as heretics, if they continued to preach and teach their beliefs, they would become rebels.