Categorification and K-theory

In this talk, I will explain and motivate the concept of categorification and present various examples. The Euler characteristic is an invariant of a topological space, that serves as a shadow of a more refined category theoretic invariant—homology—which retains significantly more information. The existence of such a categorical construction underlying a numerical one is a common phenomenon in topology and algebra. I will also discuss Khovanov's question on the existence of categorification of arbitrary rings.