This might be a large ask, so I'll preface it with my own current (and not-so-thorough) understanding. From what I've gathered, the strength of the Catholic Papacy and its grip over European leaders in the 12th-13th centuries enabled the likes of the Reconquista to continue, the Albigensian Crusades, and the far-and-away excursions into the Levant. My interest lies in the ramifications of such a Papacy's absence or, at the very least, toned down influence. What effects would it have if, for instance, Raymond of Toulouse and Peter II of Aragon were allowed to continue their secular policies, whether through their protection of Cathars or the inclusion of Jews within the bureaucracy. Or if Sicily continued to be a bastion of secularism? Or if the expulsion of the Jews in England under Edward I and in Iberia never happened? All of this depends on if such a state of affairs is even possible, and I realise I opened up multiple cans of worms, so I suppose greater experts can poke holes here.