ScalingStacks

Remark 7.4.23. We leave to the reader to check the following alternate definition of the product in the strand category.

We have $\theta^{\prime}\cdot\theta\neq 0$ if and only if there are parametrized braids $\vartheta,\vartheta^{\prime}$ with $\theta=[\vartheta]$ and $\theta^{\prime}=[\vartheta^{\prime}]$ and there are $\alpha:I^{\prime}\to I$ and $\alpha^{\prime}:K\to K^{\prime}$ two parametrized braids with $I^{\prime},K^{\prime}\subset Z\setminus Z_{exc}$ such that $i(\alpha)=i(\alpha^{\prime})=0$, $\alpha^{\prime}_{\vartheta^{\prime}\circ\vartheta\circ\alpha_{s}(1)}\circ\vartheta^{\prime}_{\vartheta\circ\alpha_{s}(1)}\circ\vartheta_{\alpha_{s}(1)}\circ\alpha_{s}$ is admissible for all $s\in I$ and $i(\alpha^{\prime}\circ\theta^{\prime}\circ\theta\circ\alpha)=i(\alpha^{\prime}\circ\theta^{\prime})+i(\theta\circ\alpha)$.