> This is not written anywhere explicitly in the docs.

It absolutely is. I'll use the Elixir docs as my source:

> A non-temporary child process may later be restarted by the supervisor.

And, further up in the docs when talking about the circumstances under which a supervisor will restart a child that has terminated: [0]

Restart values (:restart)

The :restart option controls what the supervisor should consider to be a
successful termination or not. If the termination is successful, the
supervisor won't restart the child. If the child process crashed, the
supervisor will start a new one.

The following restart values are supported in the :restart option:

:permanent - the child process is always restarted.

:temporary - the child process is never restarted, regardless of the
supervision strategy: any termination (even abnormal) is considered
successful.

:transient - the child process is restarted only if it terminates
abnormally, i.e., with an exit reason other than :normal, :shutdown, or
{:shutdown, term}.

For a more complete understanding of the exit reasons and their impact, see
the "Exit reasons and restarts" section.

And the "Exit reasons and restarts" section says: [1]

> A supervisor restarts a child process depending on its :restart configuration. For example, when :restart is set to :transient, the supervisor does not restart the child in case it exits with reason :normal, :shutdown or {:shutdown, term}.

You go on to say:

> But here I have synchronous function [to affect the state of a supervisor] with no indication or warnings that it’s a message.

Before I get into that, I have two questions for you:

1) How do you affect an Erlang or Elixir process without sending it a message? The docs for Processes [2] don't indicate any other way.

2) Have you never seen or written a function that does not return until it receives the response to an async operation?

Continuing on... from the top of the Supervisor docs, we see:

> A supervisor is a process which supervises other processes, which we refer to as child processes.

"A supervisor is a process...", straight off the bat. That's super clear and explicit, but I'll keep walking through the docs to show you how else this information is communicated to the reader.

If we read on, we see that the first argument to the 'stop_child/2' and 'delete_child/2' functions is of type 'supervisor()', which is defined as '@type supervisor() :: pid() | name() | {atom(), node()}'. What are these? Well, check the docs for how you start a Supervisor. [3] They say three interesting things:

1) The second argument to 'start_link/2' is of type 'option()', which is defined as '{:name, name()}', and 'name()' is defined as 'atom() | {:global, term()} | {:via, module(), term()}' . Keep those types in mind.

2) "If the supervisor and all child processes are successfully spawned (if the start function of each child process returns {:ok, child}, {:ok, child, info}, or :ignore), this function returns {:ok, pid}, where pid is the PID of the supervisor. If the supervisor is given a name and a process with the specified name already exists, the function returns {:error, {:already_started, pid}}, where pid is the PID of that process."

Notice how often it talks about "spawning" the supervisor and returning a PID, and saying that that PID is the PID of the supervisor you just created, or of a named supervisor that already exists.

3) "The options can also be used to register a supervisor name. The supported values are described under the "Name registration" section in the GenServer module docs."

Let's look at the "Name registration" section. [4] I'm not going to quote the whole thing because it'd be a nightmare to reformat sensibly, but the two key sections are

> Both start_link/3 and start/3 support the GenServer to register a name on start via the :name option. Registered names are also automatically cleaned up on termination. The supported values are: an atom ... {:global, term} ... {:via, module, term}...

and the last four items in the bulleted list in the section beginning with

> Once the server is started, the remaining functions in this module (call/3, cast/2, and friends) will also accept an atom, or any {:global, ...} or {:via, ...} tuples. In general, the following formats are supported:

Notice how those bullets match up to the 'name()' type that is passed in to supervisor:start_link/2, and connect that information with the fact that the docs for that function direct you here to learn about how you can register a name for your supervisor. Combine that information with the fact that the first argument to the "Tell the supervisor to do something" functions is of type 'supervisor()' and the fact that 'start_link' returns a PID, and it's really, really clear that a supervisor is another process that you can (optionally) name and refer to by name, rather than PID.

Once we understand that a supervisor is a process, and that the functions to instruct a supervisor to do things require the information required to contact a process, what other conclusion can we draw than "Communications with a supervisor is async, because communications with all processes are async."?

[0] <https://hexdocs.pm/elixir/1.18.3/Supervisor.html#module-rest...>

[1] <https://hexdocs.pm/elixir/1.18.3/Supervisor.html#module-exit...>

[2] <https://hexdocs.pm/elixir/1.18.3/processes.html>

[3] <https://hexdocs.pm/elixir/1.18.3/Supervisor.html#start_link/...>

[4] <https://hexdocs.pm/elixir/1.18.3/GenServer.html#module-name-...>