I like one of the simplest and most well known examples: the category of finite sets and bijections categorifies the natural numbers. Or rather it un-de-categorifies the de-categorification that led to much of mathematics in the first place. That makes it pretty special, even if it is rather basic compared with other examples.

I like one of the simplest and most well known examples: the category of finite sets and bijections functions (see below for comments) categorifies the natural numbers. Or rather it un-de-categorifies the de-categorification that led to much of mathematics in the first place. That makes it pretty special, even if it is rather basic compared with other examples.