Tensor product algebras, Grassmannians and Khovanov homology

We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of $(\mathbb{C}^2)^{\otimes \ell}$, and admitting a direct generalization to other Lie algebras. While the construction discussed is a special case of that given in previous work of the author, this paper contains new results about the special case of $\mathfrak{sl}_2$ showing an explicit connection to Bar-Natan's approach to Khovanov homology, to the geometry of Grassmannians, and to the categorified Jones-Wenzl projectors of Cooper and Krushkal. In particular, we show that the colored Jones homology defined by our approach coincides with that of Cooper and Krushkal.