# Why is Khovanov homology considered a 'categorification'?

I understand that the Euler characteristic of Khovanov homology is the Jones polynomial. But in what sense does this give category theory structure to the Jones polynomial, i.e., what are the objects and morphisms?

• Categorification is not really meant in that sense. Rather it means that while the Jones polynomial is an element of a set, namely the set of polynomials, the Khovanov homology is an object of a category, namely the category of graded abelian groups (or, better, some derived category of chain complexes). – Mike Shulman Jun 18 '17 at 14:48

Khovanov (2002) sets this up in a "tangle 2-category:" the objects are sets of even numbers of oriented points. The 1-morphisms are planar diagrams of cobordisms between them (so that which line in a crossing is on top is important). Hence a 1-morphism is a (diagram of a) tangle, and 2-morphisms are diagrams of oriented tangle cobordisms. Khovanov constructed Khovanov homology as a 2-functor from this category to a category built out of chain complexes of abelian groups. There are subtleties to this setup, however: Jacobsson (2003) showed that it's not functorial when generalized to $\mathbb Z[c]$-modules.