Great question.

Islam has historically had political and legislative dimensions alongside the spiritual ones. In this case, the Caliph, (Khalifa in Arabic), which means "successor" or "steward"/"deputy", is a temporal leader. The reason they are a "successor" is because they inherited the temporal seat of power of the Prophet, but not their prophethood, which is an important distinction. This means the Caliph was in charge, as a political and temporal leader, of the Islamic state that the Prophet built in his struggle against the Quraishi persecution, but the Caliph had no say in the actual doctrine of Islam. They could declare something but in the theology of Islam that didn't make it doctrinally true, and traditionally the Caliphs throughout history rarely attempted to do so because it went against the entire point of the din, it would have probably evoked revolutions.

In terms of leadership, there really isn't one. Let's approach this question from the perspective of an average Muslim. Who, as a Muslim, do you learn about Islam from? What institutions do you approach to practice your faith?

Well, first, there is the mosque. The mosque is run by an Imam. What makes an Imam an Imam? Their job description is, essentially, the caretaker of the mosque, and the person who leads prayers. And what makes a mosque a mosque is if Muslims congregate there. There is no ritual where the imams are holy ordained by a higher authority, and you don't need to ask any specific group for permission to build a mosque. Theoretically, anyone could be an Imam right now, by saying their house is a mosque and saying they are its imam. Nobody can deny this claim on a spiritual basis. They can argue no one uses it but that just means you have to attract a following. In this way, the position of Imam and the legitimacy of a mosque develops organically depending on whether the community decides to use it as such.

Of course, there are other positions that people divest prestige in. Sheiks, muftis, mullahs, etc. These are essentially just learned people. Their word is not automatically correct or holy. Mullah, for example, usually implies they attained a degree in one of the Islamc studies. A Mufti is more specialized in sharia, the legal dimension of Islam. Again, it just refers to the fact that they had a degree in it. Sheik doesn't even have anything like that, you become a Sheik based on enough people calling you that as a form of respect because you seem so knowledgeable. The "Ulama" is a learned class of Islamic scholars. There are some Ulamas given State authority in their countries in the modern day, but the Islamic viewpoint traditionally has dissuaded this kind of behavior.

None of these people have authority on Islam, either. They can issue "fatwas" - that is, essentially, legal advice based on their interpretation of scripture - but once again, there is nothing that gives their word any higher authority than anyone else's other than the implication that they studied it. For every fatwa a mufti has issued, you can find a fatwa from a different mufti that has the exact opposite opinion. Usually, neither of them would consider the other heretical, nor would they consider you heretical for not listening to them. The ones who do, most Muslims consider them cranks.

The institutions of Islam, then, are not divested authoratively from one central source. Some countries such as Saudi Arabia attempt to use State power to make one opinion higher than another, but these are outliers - most Muslims, in most countries, and for most of their history, have had the view that Islamic doctrine is up to interpretation by everyone, and that any one person's interpretation could equally be the valid or the correct one. Of course, that does not mean we have to listen to your interpretation, but it does mean I have no authority spiritually over yours - as in, I am not ordained in any holy way to say that you are wrong, I cannot interpret God's will.

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