SUNY Potsdam, June 4, 2003.

**Abstract: ** The minor advantage of homology theory over the
Euler characteristic is that it is a finer invariant. The major
advantage is that it is a functor: Given a map between spaces there is
a map between their homologies. Think of almost any major theorem in
algebraic topology and you'll find that the functoriallity of homology
is deeply involved. In my talk, I will explain in elementary terms what
seems to be the corresponding property of Khovanov's homology: that it
is a functor from the category of links and cobordisms to the category
of vector spaces (see Jacobsson's arXiv:math.GT/0206303
and Khovanov's arXiv:math.QA/0207264).
My proof of this property is in the spirit of Khovanov's, but it is
both simpler and more general. It involves the extension of the theory
to the canopoly of tangle cobordisms, with values in several related
canopolies.

What's a canopoly? No, that would go in the talk; not here. It's an object with a rather messy formal definition but a very simple visual image.

**Handout: ** NewHandout.pdf.