If someone accuses you of stealing their lunch and you give a categorical denial, it means that you absolutely deny having anything to do with the theft. Categorical means absolute, unqualified, unconditional.
If you ask someone to marry you and she says maybe, you might be able to persuade her. If it's no, you might still have a chance. But if she gives you a categorical no, she will never change her mind. Less often, categorical is used to describe something that is part of a certain category or group. A Doberman at the Westminster Dog Show might be the categorical winner, meaning it is the best Doberman but not the winner of the show.
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