My argument isn't that Weber's definition is correct or accurate; that's a whole 'nother discussion.
It's that, to the level of a thought-stopping cliche, the definition is misstated with the emphasis on violence rather than legitimacy, which misses the whole point.
But Weber's claim is also nuanced:
- Non-government violence, if tolerated by the government, is then sanctioned by the government, and hence, government retains its monopoly on legitimacy. Again, the relevant monopoly is legitimacy. Not on who is acting with violence, but who finds that violence legitimate or illegitimate.
- Governmental violence, if deemed illegitimate by the population, means that the government no longer has legitimacy itself, and hence by Weber's definition is no longer a government (at least by his terms).
- If some other entity holds legitimacy over violence within a given area (say, a neighboring state, a local warlord, Great Power, partisan or resistance movement, corporate or commercial entity such as, say, the British East India Company), then that entity is effectively the government, again, by Weber's definition. As an example, Mohandas Gandhi's ahimsa movement successfully challenged British imperial rule not by claiming violence for itself, but by successfully claiming the principle of nonviolence. The Empire was delegitimated in the process.
- And finally, if no one institution can successfully claim legitimacy of violence within a region, then there is no government. The region is effectively stateless.
There may well be other constructs, situations may be fluid (changing with time or over space), effective control units may be small (city-states, tribes), etc.., but you can generally find a Weberian projection in such cases.
Again, I've raised this point numerous times on HN, largely due to the widespread misrepresentation of Weber. Often, I suspect, by people who have no idea that they're employing a corrupted version of his definition in the first place. I encourage you to look through my earlier discussions to see if your further objections aren't already addressed there: https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...>.
Prior discussion points to where the misleading restatement seems to have originated, more on the elements of Weber's definition as applied to various situations, Weber's original work, and translation into English which didn't occur until the 1950s, followed shortly by the bastardised variant taking hold.
I also strongly recommend British political historian David Runciman's own views on this definition, which I'd first encountered well after forming my own. They're expressed in this episode of his podcast Talking Politics: https://play.acast.com/s/history-of-ideas/weberonleadership> (at about 15 minutes).