Inbox is a reinvention of email from Google

so it's like wave, but implemented on top of normal email?

Much larger discussion found here: https://news.ycombinator.com/item?id=8493619

It may answer your question

Mailbox-style email triaging with Google Now machine learning and auto-organization. It's very powerful, because it pulls out important information in the email and displays them at the inbox level so you can, for example, reply to calendar invites without opening the email.