ScalingStacks

We consider as in §7.4.3 a family $M=\{a_{1},\ldots,a_{n}\}$ of points on $S^{1}$ and $z\in S^{1}-M$ such that $a_{1},\ldots,a_{n},z$ is cyclically ordered.

The next proposition follows immediately from Proposition 7.4.18 and Lemmas 7.4.19 and 7.4.20.

The isomorphism $\Gamma_{[1,n]^{+}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Gamma_{M}(\vec{S^{1}})$ of Lemma 7.4.19 restricts to an isomorphism of groups $\Gamma^{f}_{[1,n]^{+}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Gamma_{M}(\vec{I})$ and the isomorphism ${\mathcal{H}}_{n}^{f++}\to{\mathcal{S}}^{\bullet}_{M}(\vec{I})$ of Proposition 7.4.33 is compatible with the grading by those groups.

Consider $Z={\mathbf{R}}_{>0}$ as an unoriented curve. We denote by ${\mathcal{S}}_{\otimes}^{\bullet}({\mathbf{R}}_{>0})$ the full subcategory of ${\mathcal{S}}^{\bullet}({\mathbf{R}}_{>0})$ with objects the subsets of the form $\{1,\ldots,n\}$ for some $n\in{\mathbf{Z}}_{\geq 0}$. We define a monoidal structure on the differential pointed category ${\mathcal{S}}_{\otimes}^{\bullet}({\mathbf{R}}_{>0})$ by $\{1,\ldots,n\}\otimes\{1,\ldots,m\}=\{1,\ldots,n+m\}$ and $\theta^{\prime\prime}=\theta\otimes\theta^{\prime}$ is defined by $\theta^{\prime\prime}_{i}=\theta_{i}$ if $i\leq n$ and $\theta^{\prime\prime}_{i}=\theta^{\prime}_{i-n}$ otherwise.

The next theorem follows immediately from Proposition 7.4.33.