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We give a purely combinatorial construction of colored $\mathfrak{sl}_n$ linkhomology. The invariant takes values in a 2-category where 2-morphisms aregiven by foams, singular cobordisms between $\mathfrak{sl}_n$ webs; applying a(TQFT-like) representable functor recovers (colored) Khovanov-Rozanskyhomology. Novel features of the theory include the introduction of `enhanced'foam facets which fix sign issues associated with the original matrixfactorization 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.

The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality