ScalingStacks

Definition 2.5. The Khovanov–Rozansky $\mathfrak{gl}_{N}$ link homology $\mathrm{KhR}_{N}\colon{\mathbb{R}}^{3}\boldsymbol{\mathrm{Link}}^{\circ}\to\mathrm{gr}^{{\mathbb{Z}}\times{\mathbb{Z}}}\boldsymbol{\mathrm{AbGrp}}$, with target given by the category of ${\mathbb{Z}}\times{\mathbb{Z}}$-graded abelian groups and homogeneous homomorphisms, is defined as the composition of $\left\llbracket-\right\rrbracket$, the representable functor $\bigoplus_{k\in{\mathbb{Z}}}\boldsymbol{\mathrm{Foam}}_{N}(q^{-k}\emptyset,-)$, and taking homology. It is functorial by Theorem 2.4.