ScalingStacks

When the lasagna filling $G$ consists of a surface $S$ (an embedded cobordism $S\subset Z$ from $L$ to $L_{1}$) with no input balls, we will simply write $\Psi_{Z;S,\alpha}$ for $\Psi_{Z;G,\alpha}$. Furthermore, we could decorate $S$ with $n$ dots at a chosen location, for $0\leq n\leq N-1$, as usual in $\mathfrak{gl}_{N}$ foams; cf. [27, Example 2.3]. This corresponds to constructing a lasagna filling $S(n\bullet)$ with $n$ input balls intersecting $S$ along unknots, each decorated with the generator

$$X\in\operatorname{KhR}_{N}(U)\cong{\mathbb{Z}}[X]/(X^{N}).$$

(This filling is equivalent to one where we consider a single input ball intersecting $S$ in an unknot, decorated with $X^{n}$.) When the chosen location of the dot placement is clear from the context, then we denote the corresponding map by

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$$\Psi_{Z;S(n\bullet),\alpha}:\mathcal{S}_{0}^{N}(W;L,\alpha)\to\mathcal{S}_{0}^{N}(W\cup Z;L_{1},\alpha_{1}).$$