In this paper we show how the colored Khovanov-Rozansky Sl(N)-matrix factorizations, due to Wu [45] and Y.Y. [46,47], can be used to categorify the type A quantum skew Howe duality defined by Cautis, Kamnitzer and Morrison in [14]. In particular, we define Sl(N)-web categories and 2-representations of Khovanov and Lauda's categorical quantum sl(m) on them. We also show that this implies that each such web category is equivalent to the category of finite-dimensional graded projective modules over a certain type A cyclotomic KLR-algebra. (C) 2018 Elsevier B.V. All rights reserved.

We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n, r) for 3 <= r < n, using results on diagrammatic categorification in affine type A by Elias-Williamson, that extend the work of Elias-Khovanov for finite type A, and Khovanov-Lauda respectively. We also define 2-representations of these categorifications on an extension of the 2-category of affine (singular) Soergel bimodules. These results are the affine analogue of the results in [28].

This is a follow-up to the paper in which we categorified the affine quantum
Schur algebra S(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's
categorification of the affine quantum sl_n. In this paper we categorify S(n,n)
for n > 2, using an extension of the aforementioned quotient.