There are specific items that have to be met in order to be a part of the Olympics. Some of the key requirements:
The Korean government spent a lot of time codifying Taekwondo. They created the rules and sponsored instructors and new schools throughout the world.
The World Karate Federation has taken steps to join the Olympic Games. In 2005, they made a bid to be included in the 2012 London Games, but they were not successful. In 2009, they made another bid to be included in the 2016 Rio Games, but they were not successful.
One of the obstacles to creating new Olympic sports is related to the decision by the International Olympic Committee (IOC) to limit the size of the games. Currently, the summer Olympics is limited to 10,500 athletes. In order to add a sport, another sport has to give up spaces for athletes. This same obstacle makes it difficult to add the women's version of some sports.
At the 2008 Beijing Games, taekwondo was limited to 128 athletes: 64 men and 64 women. No team may bring more than 4 athletes. Instead of the traditional. 16 weight classes. Olympic taekwondo is limited to 8 weight classes.
Why don't they put some indoor sports on the winter olympic schedule?
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The primary reason is that karate is not a sport. It is a martial art designed for use in combat.
The other reason is the lack of a single ruling body to organize and set rules internationally. There are hundreds of styles and dozens of organizations, and they do not like to cooperate with each other and 'share' secrets.
And it is very dangerous. You see taekwondo and judo in the Olympics because they have been highly codified with very careful rules under a single international ruling body.