User toby bartels - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:16:47Z http://mathoverflow.net/feeds/user/1702 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4841/what-precisely-is-categorification/5094#5094 Answer by Toby Bartels for What precisely Is "Categorification"? Toby Bartels 2009-11-11T19:46:23Z 2009-11-11T19:46:23Z <p>As a proponent of <a href="http://ncatlab.org/nlab/negative+thinking" rel="nofollow">negative thinking</a>, instead of saying that categorification ‘replaces sets by categories’ (to quote Wikipedia), I would say that we replace truth values by sets, especially the truth values of equations. That is, we acknowledge that there may be many different ways in which something may be true, in particular many different ways in which two things may be the same. And then it is meaningful to ask whether two ways in which these things are the same are the same way (and if so, whether two ways that <em>they</em> are the same are the same way, etc).</p> <p>In particular, while two elements of a set simply may (or may not) be equal, two objects of a groupoid may be isomorphic in many different ways. And while two parallel isomorphisms in a groupoid may be equal, two parallel equivalences in a \$2\$-groupoid may be isomorphic in many different ways. Or while one element \$x\$ of a poset may precede an element \$y\$, there may be many different morphisms from one object \$x\$ of a category to an object \$y\$.</p> <p>As you can see from these examples, I would distinguish categorification proper from the possibility of adding noninvertible arrows (which I would call ‘laxification’). Often one categorifies and then laxifies, but often one only categorifies.</p>