Limitations (?) to Swift extensions.

Async.swift

class Async<Wrapped> {
  
  func map(function: @escaping (Wrapped) -> NewWrapped) -> Async<NewWrapped> {
    // ...
  }       

  // ...
}

typealias AsyncResult<Success, Failure> = Async<Result<Success, Failure>>

extension AsyncResult {

  // The function below cannot be declared because there are no Success or Failure types in scope
  func mapSuccess<NewSuccess>(function: @escaping (Success) -> NewSuccess) -> AsyncResult<NewSuccess, Failure> {
    // ...
  }
  
  // ...
}

I think you can get what you're after without the typealias by adding a where clause to your extension function:

extension AsyncResult {
  func mapSuccess<New>(function: @escaping (Success) -> New) -> AsyncResult<New, Failure> where Success == Result<New, Failure> {
    // ...
  }
}

You can constrain the function's generic New or the instance's generics Success, Failure or both.