The academic term for this is monopoly on force. It means the military can use force in the city, such as shooting people, taking people captive, threatening people, etc. without receiving any punishment from others. Note that individual soldiers may get punished for not following the orders and rules of the military they are in but the military as a whole have a monopoly on force in the city.
This is complex to explain as ELI5… The question is twofold, and I think you mean both the administrative and the military side of it. First you should ask yourself what it means for a governmening body to have administrative control of a city/town/village. It means that they have established control over the population in the area, who are then compelled into following their rules. This is mostly done through passing and enforcing laws.
So how does one make sure that the citizens follow these laws? Well, violence. You break a law, you go to jail. You are forcibly stripped of your freedom and other rights. Because of this necessary mechanism, the state needs to be the only entity allowed to inflict violence upon its population. You can’t have other organizations or individuals causing violence – and enforcing their rules through violent coercion. The other third party “brokers of violence”, so to speak, must not be allowed to use violence inside the borders. It’s why you see mall security or private security firms calling the cops – they can’t be allowed to enforce their own rules on the population, they must go through the legal framework for violence.
Now that we’re done with this lengthy context, what does an invading force need to do to control an area? Well, they have to eliminate the threat of violence from the old governing body and enforce laws through their own newfound monopoly on violence. All you have to do is force the local military out(the combat force that can realistically resist violence), take over government buildings and either oust the old officials and put new ones in place, or force them to comply to your new laws through the use of violence.
You don’t even necessarily need an old governing body to take over. You can just employ violence on the local population from the get-go and coerce them into compliance that way. Parade a few tanks through town, shoot a few detractors, make sure key locations stay occupied by your forces. And then just handle administrative business with your own officials in charge.
As for the military capture side of it, it’s a bit more vague but it just means the local army has been forced to withdraw completely from the combat zone and the invading army is now entrenching defensive positions, and working on administrative control over the area to force the local population to collaborate with the invasion, such that they can set up outposts safely to conduct logistical and command operations.
A city is normally run by a local government with some elected combination of councilors, mayors, and executives. They appoint into various positions around the city people who run the city from gardeners to head’s of the sports department.
When a military takes control of a city they have essentially taken on the job of being the elected and non-elected components of the city. They may decided that tree watering won’t happen under their watch. They may dissolve the local police force…. or absorb it into their own structure.
Imagine there’s a playground full of kids, and one big kid is making sure that everyone is playing nice with each other, being safe, etc. But, everyone has to play by the big kid’s rules or he’ll make them go home. Now, one day a big bully comes along and beats up the big kid, and makes him run home. Now the bully is the big kid on the playground, and everyone has to play by his rules.
It’s kinda like that. In the most basic sense, taking control of *any* strategic location, including a city simply means having troops in that location and the ability to exert a level of control over that location that would otherwise belong to the opposing side.
It means that their forces occupy the city: they live there, patrol the streets, regulate the people, set up defenses etc. Essentially, it means that non-aligned forces cannot operate in the area.
Think of it like a house. If I’m upstairs and my enemy is downstairs, and he fights me any time I try to go downstairs, neither of us can be said to have control over the house. I can’t go downstairs and do stuff until he’s gone.
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