Here's why:
* It's a sport: Badminton is the name of the sport itself, not a particular team, player, or tournament.
* Proper nouns are specific: Proper nouns are capitalized because they refer to unique things. Badminton is a general term for a sport.
Examples of proper nouns related to badminton:
* The All England Open Badminton Championships: A specific tournament.
* Viktor Axelsen: A specific badminton player.
* Yonex: A specific badminton equipment brand.