  In this paper we use colored sl(N)-matrix factorizations, due to Wu and Y.Y.,
in order to categorify part of the quantum skew Howe duality defined by Cautis,
Kamnitzer and Morrison. In particular, we define web categories and
2-representations of Khovanov and Lauda's categorical quantum sl(m) on them. We
show that each such web category is equivalent to the category of finite
dimensional graded projective modules over a certain level N cyclotomic
Khovanov-Lauda-Rouquier algebra.
