Category denial is a rhetorical tactic used by people with a strong ulterior motive to dislike whomever or whatever they are speaking against. It attempts to leave the boundaries of the argument, reframing the debate in a new lens more sympathetic to the speaker's goal. This is accomplished by implying the target concept, typically but not always what is being attacked by the speaker, fails to meet the requirements for the category, and since the debate presupposes a certain categorical domain, any defence (or whatever the opposition is doing) so far constructed of the concept is invalid. In this sense, it is like transcendentally moving the goalposts.
An example: while in a town centre, I saw a police notice up on the town bulletin board. It read "Using dogs to hunt hares is not a sport. It's a crime." Now, what is a sport? Obviously, being reasonable, you can't have a definition of sport that both includes things like football, tennis, horse racing, boxing and so on, and excludes hunting. What the council over this town tried to do is to deny hare hunting its place in the category of sports so as to deny its advocates the opportunity to defend it on those grounds (which I assume is something they would do). Instead, they have placed it in the category of crimes, members of which are of course much harder to defend.
In its most widespread variant, category denial takes the form of "not even (...)". Oftentimes you may see the phrase "not even wrong". This is a little more subtle than the example in the last paragraph, because the important point is not that it's trying to deny, let's say X, it's place in the category of wrong things, but in the category of truth claims. If you say a truth claim is wrong, you are going to be asked to show that. However, if you say that the claim is not even wrong, you aren't bound to even debate it at all because you've said X is not even capable of being proven right or wrong. (Usually I think this isn't used to such a degree, and even more rarely is it used to intentionally have this effect, which is a possibility for all sophistries.)
In this manner you would completely sidestep actually having to argue you are correct, should your opponent let you get away with it. In this case, the idea that something is not even wrong is getting across the idea that X is either:
-
based on fundamentally faulty suppositions
(bad axioms if you like), or
headed in the wrong direction entirely, which is
to say it may actually be right or wrong but has no
bearing on the discussion at hand.
When facing category denial, the first thing is to check if the person using it is right. If they are, there's not much you can do but accept it. Most of the time though, they are using it sophistically and aren't right, and in this case the simplest way to refute the tactic on the spot is to refuse it. In the last example, for example, you could just ask what is meant by "not even wrong" and the answers would reveal that they really just used a nonsense phrase to make an impression. So coming up against (for the first real example) "hate speech is not speech" is as easy as just refusing the attempted shift, which in this case is accomplished by asking what speech is. They can't answer the question and stay consistent without some horribly convoluted definition of, in this case, speech that even they probably don't believe. Most of the time you encounter this you won't be actually in debate with the user, so it's enough to just notice it and recognise it's fallacious.