Here's how I think about this question abstractly, as it pertains to «membranes/boundaries».

This is only meant to be an abstract post— but soon I will publish the first post of a sequence that applies this in everyday life as rationality technique.

Epistemic status: I'm a bit uncertain about what I've said in this post, and I'm looking for feedback. I think there's definitely something real in what I've said here, though.

Update: I have a cleaner and better model coming soon

What's inside an agent's «membrane»

Agents are formed by their boundaries/membranes. If they did not maintain homeostasis of their boundaries, they would have died long ago. And they must continue to maintain their boundaries (ie: keep bad things out & keep sensitive things in) if they are to self-perpetuate. 

I think the things that are in an agent's boundary/membrane can be most directly summarized by "the things that are within that agent's control" / sovereignty / autonomy / self-governance… Moreover, the things that no one else will do (that also aren't part of another agent's «membrane»).

[Also, if it belongs to one agent, it doesn't belong to any other agent.]

[In this post, I'm going to ignore boring/obvious considerations of "my skin is a membrane and my other organs are inside" and instead mostly discuss nonphysical things.]

I like to break this up into 4 categories. I might think differently about this in the future, but this is the way I think about it now. 

I will take myself as an example. Things that are "in my boundary/membrane"… 

1. Things they alone observe

I alone observe my subjective experience.

• Notably, my feelings
  • (This is also why it can be triggering for someone to tell you how you feel.)
• No one else is able to (directly) observe these things.

2. Things they alone control

I alone control my actions.

• Note that I mean control in a Cartesian Frame sense: "If $S$ is both ensurable and preventable in $C$, we say that $S$ is controllable in $C$." Ie: If you can both (totally) ensure and (totally) prevent some choice, then you control it.
  • Thus, control is different from mere influence.^[1] E.g. I may be able to influence your decisions, but I don't control them.

3. Things they alone have to do

I.e.: Responsibility for my preferences

• E.g. 1: I have a body, and I want to live, and I also control my body, so I am responsible for my own hunger. In general, I am responsible for working towards my own desires. Who else is going to manage these things?
• E.g. 2: I legally own and therefore legally control my house, and I want there to be electricity in my house, so I’m responsible for paying my electric bill. Who else is going to manage that?
• Also, I want to be clear with this. Here are examples of things that are not in my boundary despite #3:
  • E.g. -1: Making a girl like me back when I want her to
    • But I can't control her. (She has her own «membrane»/sovereignty!)
  • E.g. -2: Making someone do a thing
    • But I can't control them. (They have their own «membrane»/sovereignty!)

Note: I'm uncertain that I've got this section totally right. I might revise this later.

4. Things others have given them permission for (that they've accepted)

Things other people give me permission to control — via agreement or a formal or informal contract.

• E.g. 1: I have a bank account. I live in a society that enforces ownership of bank accounts. There are contracts (and in this case, laws) that organize this. And there are systems to enforce these contracts.
• E.g. 2: I own my house, because I have a contract with the government (and perhaps a bank) that says this. And the contract would be enforced if someone tried to do anything that requires ownership of my house.
• E.g. 3: Alex is employed by a company as their office manager. For him to do his job well, they must give him permission for property and responsibilities. 
  • Property like keys, a desk, passwords. 
  • Responsibilities like keeping the fridge stocked. 
    • For the office to give him responsibility means the company is opening a boundary to him in the necessary specific ways— and essentially this means that they won't take action if he meddles with e.g. the organization of the supply closet. (Because that's the job they've given him.)
      • Meanwhile, if anyone else who isn't their office manager did such a thing, the company would (should^[2]) take action against that person. 
• In a way, when we make a contract with someone, they're giving us a piece of themselves (or we're giving them a piece of ourselves). The giving party could also revoke this at any time, however, and the receiving party could reject the offer at any time.

Notice the symmetry of this section with #3. (Or, maybe #4 might better well be considered part of #3?)

[See the final section of this post for more about contracts.]

And nothing else!

Everything else in the universe— everything I do not explicitly have within my boundary— is not within my boundary/membrane.

• That is, by default, any particular object or responsibility is not within my boundary/membrane. By default: every house isn’t mine, every unpaid electric bill isn’t mine, every body isn't mine, every experience isn't mine. (How could it be any other way?) 
  • And in general if I see any object or responsibility in the world, I assume that it’s not mine and thus not mine to manage. 
  • So this forms my prior: “Assume that I don’t own X for all X.” Then, on top of this "nothing is in my boundary/membrane" prior, I add the things that are obviously mine— things I control, things I observe, things I want to happen, things I have permission for. These are the things that are within my boundary/membrane.

more about contracts

Critch writes something similar in «Boundaries», Part 3b:

your electrician is allowed to affect your home, and other aspects of your life in general, if they do so via electrical repairs on your home that you've consented to. Thus, by default you yourself serve as a boundary between your home and your electrician, and when you open up that boundary for the purpose of electrical repairs, the repairs on your home are supposed to be a boundary between your electrician and the other aspects of your life. 

(This also relates to Critch’s Part 3a: Nonviolent boundary crossings.)

I.e.: For electrical repairs, you have allowed an electrician access to your home only for the specific and bounded purpose of electrical repairs. There is an agreement/ (formal or informal) contract. This is what I mean.

'Boundary/membrane transactions' occur via mutual agreements/contracts. 

Essentially: "I agree to give you responsibility for (X : something that is mine) under conditions Y" and simultaneously "I agree to accept responsibility for (X : something that is yours) under conditions Z".

Note: I explain a detailed example of this in this post.

1. ^{^}

   “Control” is different from mere influence. As long as I respect your membrane/boundary, I can only influence you if you give me permission— i.e.: if you open up your boundary/membrane to me in this way.

   E.g.: I can’t control what you eat. And by default I can’t influence what you eat. But you can give me permission to influence you, eg you can ask me for advice on what you order for dinner.

   Theoretically, I could try to control what you eat in that example by paralyzing and intubating you, but I still can't control that I'll be successful. For two, this would also be clearly violating your boundary/membrane. And third of all, are you even the same person after I do that? Because I basically just murdered you and replaced you with a lobotomized clone.

2. ^{^}

   The company has to do this or else it will lose control of something that is within its own membrane/boundary. (For the company legally owns/rents the office.)