I am create a firebase based chat application for iOS. One of the features is to send push notification to the users in the same chat room (if they are offline).
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Displaying alert badges and notifications on iPhone applications is accomplished through Apple's Push Notification system. Since the application is not running on the user's phone when they receive notifications, the APN will have to be triggered from your server-side code.
You'll probably want to create a server-side module that listens for changes to your chat Firebase. When a message appears for a user that is offline, you'll have to schedule a remote notification with the APN. That latter part has nothing to do with Firebase, but has extensive documentation on Apple's developer web site.
I'm not sure if a web application can display alerts or badges. Otherwise this approach will only work if you create a native wrapper for your Firebase chat application.