Now that many professionals athletes are taking part in the Olympic Games, why do pro boxers not also compete in the boxing competition?
Are there any other categories that do not take part in the Olympic Games?
The governing body of Olympic boxing, the AIBA, only recently changed its rule for the 2016 Olympic Games to permit boxers to qualify. To do that, they have to be an otherwise amateur boxer who participates in an AIBA sanctioned professional tournament called the APB. As far as I can tell from the AIBA site, participation in any other pro tournament still disqualifies a boxer from competing in the Olympics, though it's a bit unclear.
Keep in mind, though, that we only have one data point to look at when we say that professional boxers don't compete in the Olympics. We'll need to see what happens in 2020 to get a better idea of how the AIBA is implementing these new rules. (Of course, right now there's a question of whether the IOC is even going to allow boxing at all, but that's a different question.)
It's also important to note that boxing, probably more than any other professional sport, is rife with political and financial motives at the expense of the athletes and the quality of the sport; my best guess is that something along those lines is going on here, with all the other professional boxing federations not wanting their boxers to participate in an AIBA tournament, boxers not wanting to forego their usual salary and risk getting seriously injured fighting for the AIBA, etc. These various reasons could have kept a lot of professionals out of the games in 2016, but with time to prepare and negotiate new contracts some of those issues may go away.