291 However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero. |
291 However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero. |
292 Consider then $$\widetilde{q} = \sum_i \left(a_i \tensor \widetilde{q_i} \tensor b_i\right) - 1 \tensor \left(\sum_i a_i \widetilde{q_i} b_i\right) \tensor 1.$$ Certainly |
292 Consider then $$\widetilde{q} = \sum_i \left(a_i \tensor \widetilde{q_i} \tensor b_i\right) - 1 \tensor \left(\sum_i a_i \widetilde{q_i} b_i\right) \tensor 1.$$ Certainly |
293 $\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. |
293 $\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. |
294 Further, |
294 Further, |
295 \begin{align*} |
295 \begin{align*} |
296 \hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}\right) b_i) \tensor 1 \\ |
296 \hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}) b_i\right) \tensor 1 \\ |
297 & = q - 0 |
297 & = q - 0 |
298 \end{align*} |
298 \end{align*} |
299 (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). |
299 (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). |
300 |
300 |
301 Similar arguments show that the functors |
301 Similar arguments show that the functors |
339 $\ev(\bdy y) = 0$, because $$C^{\tensor \ell_1} \tensor \ker(\DirectSum_k C^{\tensor k} \to C) \tensor C^{\tensor \ell_2} \subset \ker(\DirectSum_k C^{\tensor k} \to C).$$ |
339 $\ev(\bdy y) = 0$, because $$C^{\tensor \ell_1} \tensor \ker(\DirectSum_k C^{\tensor k} \to C) \tensor C^{\tensor \ell_2} \subset \ker(\DirectSum_k C^{\tensor k} \to C).$$ |
340 Similarly, if $*$ is contained in the blob, then the blob label is a sum, with the |
340 Similarly, if $*$ is contained in the blob, then the blob label is a sum, with the |
341 $j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$, |
341 $j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$, |
342 and there are labels $c_i$ at the labeled points outside the blob. |
342 and there are labels $c_i$ at the labeled points outside the blob. |
343 We know that |
343 We know that |
344 $$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \tensor \to M),$$ |
344 $$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \to M),$$ |
345 and so |
345 and so |
346 \begin{align*} |
346 \begin{align*} |
347 \ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\ |
347 \ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\ |
348 & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\ |
348 & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\ |
349 & = 0 |
349 & = 0 |