One diagram is worth a thousand words
Let me give now some more details.
One diagram is worth a thousand wordsEach step of a “categorification process” should reveal more structure. The most classical illustration of this is the following: Here we first “categorify” numbers into vector spaces. The new information available are now linear maps between vector spaces (thus, we have the whole power of linear algebra at hand). There is no reason to stop: we can “categorify” vector spaces into categories, linear maps into functors. Again, we see a new layer of information, namely the natural transformations between these functors. The last step “naturally lives” in a -categorical setup. The idea is clear: keep on going (if possible of course).
Categorification? A rough description
The idea behind categorification is, given a fixed notion one really likes, to find an “explanation” for properties of this structure by considering natural construction in a category such that the structure is some kind of shadow of these constructions. To this end, one replaces “set like structures” (i.e. -categories) with “category like structures” (i.e. -categories). Of course one can perform such a process on any level, e.g. one can categorify a “-category like structure” into a “-category like structure”. Or one can “categorify with extra structure”, e.g. categorify vector spaces instead of sets.
Or, to say it otherwise, categorification is an “inverse” process for decategorification (which is best to be defined via examples)
Some examples of decategorification are (note that the notion categorification is too new and therefore not directly used in these classical examples):
The whole idea can be summarised in the so-called “ladder of categories”.
Categorification? A rough descriptionSince categorification is a rather new subject of mathematics (well, it is not the “newest kid in the block”, but anyway), I should spend a few words about the motivation and ideas behind it. For more information, one can look at the nice introduction at n-lab link.