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Using quantum skew-Howe duality, we study the category$\operatorname{Rep}(\mathfrak{gl}(m|n))$ of tensor products of exterior powersof the standard representation of $U_q(\mathfrak{gl}(m|n))$, and prove that itis equivalent to a category of ladder diagrams modulo one extra family ofrelations. We then construct a categorification of$\operatorname{Rep}(\mathfrak{gl}(m|n))$ using the theory of foams. In the caseof $n=0$, we show that we can recover $\mathfrak{sl}(m)$ foams introduced byQueffelec and Rose to define Khovanov-Rozansky $\mathfrak{sl}(m)$ linkhomology. We also define a categorification of the monoidal category ofsymmetric powers of the standard representation of $U_q(\mathfrak{gl}(n))$,since this category can be identified with$\operatorname{Rep}(\mathfrak{gl}(0|n))$. The relations on our foams arenon-local, since the number of dots that can appear on a facet depends on theposition of the dot in the foam, rather than just on its colouring. This may berelated to Gilmore's non-local relations in Heegaard Floer knot homology.

A categorification of the skew Howe action on a representation category of $U_q(\mathfrak{gl}(m|n))$