User alex hoffnung - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:07:14Z http://mathoverflow.net/feeds/user/1671 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4841/what-precisely-is-categorification/4880#4880 Answer by Alex Hoffnung for What precisely Is "Categorification"? Alex Hoffnung 2009-11-10T16:06:57Z 2009-11-10T16:06:57Z <p>I think one important point that has been missed here is that there is not (currently) a precise answer to this question.</p> <p>There is a loose answer along the lines of that which Pete Clark gave, but I think there may be a typo in that response. And, of course, there are specific instances which shed light (and provide new mathematics) as Scott has pointed out.</p> <p>"As you can see from this article, one aspect of categorification is the systematic negation of "decategorification", which is the process of taking a category (e.g. the category of sets) and replacing it by the class of isomorphism classes of its objects (in that case, the class of cardinal numbers)."</p> <p>Categorification is NOT the systematic negation of decategorification. Decategorification can be defined in various ways as a systematic process and categorification can be understood as the non-systematic (i.e. creative) process of undoing decategorification.</p> http://mathoverflow.net/questions/4841/what-precisely-is-categorification/4880#4880 Comment by Alex Hoffnung Alex Hoffnung 2009-11-11T04:45:00Z 2009-11-11T04:45:00Z 1) Thanks for clarifying your meaning in the first issue. 2) Sorry, you are right. In fact, sheafification would be pretty bad name if it weren't idempotent. http://mathoverflow.net/questions/4841/what-precisely-is-categorification/4880#4880 Comment by Alex Hoffnung Alex Hoffnung 2009-11-11T02:16:56Z 2009-11-11T02:16:56Z On another note, what role does idempotent play in this explanation? Sheafification is not an idempotent functor for general Grothendieck topologies. It is still a systematic process. So should I read systematic as functorial?