ScalingStacks

Compatibility with the operad composition now boils down to the claim:

$$\mathrm{KhR}_{N}(D)=\mathrm{KhR}_{N}(D_{2})\circ(\mathrm{KhR}_{N}(D_{1})\otimes\mathbf{1})$$

as maps

$$\left(\bigotimes_{i\in J}\mathrm{KhR}_{N}(S_{i},L_{i})\right)\otimes\left(\bigotimes_{i\in K}\mathrm{KhR}_{N}(S_{i},L_{i})\right)\to\mathrm{KhR}_{N}(S_{2},L_{2})$$

We compare these two homomorphisms on the level of 3-ball link homologies, that is, with respect to a fixed choice of basepoints $q_{i}$ and $q$, and we suppress associators. On this level $\mathrm{KhR}_{N}(D)$ is determined by the homomorphism

(5.1)

$$\left(\bigotimes_{i\in J}\mathrm{KhR}_{N}(S_{i}^{q_{i}},L_{i})\right)\otimes\left(\bigotimes_{i\in K}\mathrm{KhR}_{N}(S_{i}^{q_{i}},L_{i})\right)\xrightarrow{T}\mathrm{KhR}_{N}(X,\sqcup_{i}L_{i})\to\mathrm{KhR}_{N}(S_{2}^{q},L_{2})$$

where $X$ denotes the boundary connect sum of the 3-balls $S_{i}^{q_{i}}$ for $i\in J\cup K$ along the tree $T$, and the first map is provided by lax monoidality. On the other hand, the homomorphism $\mathrm{KhR}_{N}(D_{2})\circ(\mathrm{KhR}_{N}(D_{1})\otimes\mathbf{1})$ is determined by the composite

	$\displaystyle\left(\bigotimes_{i\in J}\mathrm{KhR}_{N}(S_{i}^{q_{i}},L_{i})\right)\otimes\left(\bigotimes_{i\in K}\mathrm{KhR}_{N}(S_{i}^{q_{i}},L_{i})\right)\xrightarrow{T_{1}\otimes\mathbf{1}}$	$\displaystyle\;\mathrm{KhR}_{N}(X_{J},\sqcup_{i\in J}L_{i})\otimes\left(\bigotimes_{i\in K}\mathrm{KhR}_{N}(S_{i}^{q_{i}},L_{i})\right)$
	$\displaystyle\xrightarrow{\mathrm{KhR}_{N}(\Sigma^{\prime}_{1})\otimes\mathbf{1}}$	$\displaystyle\;\mathrm{KhR}_{N}(S_{1}^{q_{1}},L_{1})\otimes\left(\bigotimes_{i\in K}\mathrm{KhR}_{N}(S_{i}^{q_{i}},L_{i})\right)$
	$\displaystyle\xrightarrow{T_{2}}$	$\displaystyle\;\mathrm{KhR}_{N}(X_{K},\sqcup_{i\in\{1\}\cup K}L_{i})$
	$\displaystyle\xrightarrow{\mathrm{KhR}_{N}(\Sigma^{\prime}_{2})}$	$\displaystyle\;\mathrm{KhR}_{N}(S_{2}^{q},L_{2}).$

Here we write $X_{J}$ for the boundary connect sum of the 3-balls $S_{i}^{q_{i}}:=S_{i}\setminus\{q_{i}\}$ for $i\in J$ that is determined by $T_{1}$, and $X_{K}$ for the boundary connect sum of the $S_{i}^{q_{i}}$ for $i\in\{1\}\cup K$ along $T_{2}$. After commuting the map induced by $\Sigma^{\prime}_{1}$ past the second monoidality map, we arrive at

(5.2)

$$\left(\bigotimes_{i\in J}\mathrm{KhR}_{N}(S_{i}^{q_{i}},L_{i})\right)\otimes\left(\bigotimes_{i\in K}\mathrm{KhR}_{N}(S_{i}^{q_{i}},L_{i})\right)\xrightarrow{T}\mathrm{KhR}_{N}(X,\sqcup_{i\in J\cup K}L_{i})\xrightarrow{\mathrm{KhR}_{N}(\Sigma^{\prime}_{2}\circ(\Sigma^{\prime}_{1}\cup\mathbf{1}))}\mathrm{KhR}_{N}(S_{2}^{q},L_{2}).$$

Since the link cobordism $\Sigma^{\prime}_{2}\circ(\Sigma^{\prime}_{1}\cup\mathbf{1})$ is isotopic to $\Sigma$, the functoriality of $\mathrm{KhR}_{N}$ implies that the maps in (5.1) and (5.2) are equal. This proves the claim. ∎