ScalingStacks

In the following we will also consider enriched $2$-homs. For objects $s,t$ and $1$-morphisms $A,B\colon s\to t$ we define the bigraded $\mathbbm{k}$-modules:

(21)

$$H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(A,B):=\bigoplus_{k\in{\mathbb{Z}}}\operatorname{Hom}_{H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})}(A\{k\},B)$$

Here one grading, the quantum grading, is given by the displayed direct sum, while the other grading, the homological grading, is already internal to $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})$. Using the grading shift automorphisms, these enriched $2$-homs admit composition maps and thus assemble into a bigraded $\mathbbm{k}$-linear enriched morphism category $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(s,t)$ whose objects are the $1$-morphisms from $s$ to $t$.