  We give a purely combinatorial construction of colored $\mathfrak{sl}_n$ link
homology. The invariant takes values in a 2-category where 2-morphisms are
given by foams, singular cobordisms between $\mathfrak{sl}_n$ webs; applying a
(TQFT-like) representable functor recovers (colored) Khovanov-Rozansky
homology. Novel features of the theory include the introduction of `enhanced'
foam facets which fix sign issues associated with the original matrix
factorization formulation and the use of skew Howe duality to show that
(enhanced) closed foams can be evaluated in a completely combinatorial manner.
The latter answers a question posed in math.GT/0708.2228.
