Lemma 6.2.13 (0P81) · ScalingStacks

0P81

Lemma 6.2.13. Given $n^{\prime}\leq n$ and $\alpha:\{1,\ldots,n^{\prime}\}\hookrightarrow\{1,\ldots,n\}$ an increasing injection, the functor $F_{\alpha}$ induces a differential $\Gamma_{n}$ -graded pointed functor ${\mathcal{H}}_{n^{\prime}}\to{\mathcal{H}}_{n}$ .

0P82

Proof. Let $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ . We have $L(F_{\alpha}(\sigma))=(\alpha\times\alpha)(L(\sigma))$ , hence $\ell(F_{\alpha}(\sigma))=\ell(\sigma)$ . We have $R_{\alpha}(\llbracket\sigma\rrbracket)=\llbracket F_{\alpha}(\sigma)\rrbracket$ , hence $L_{\alpha}(m(\sigma))=m(F_{\alpha}(\sigma))$ . We deduce that $\Gamma_{\sigma}(\deg(\sigma))=\deg(F_{\alpha}(\sigma))$ .

We have $D(F_{\alpha}(\sigma))=(\alpha\times\alpha)(D(\sigma))$ and $F_{\alpha}(s_{i_{1},i_{2}})=s_{\alpha(i_{1}),\alpha(i_{2})}$ for $i_{1},i_{2}\in\tilde{I}$ with $i_{1}-i_{2}{\not\in}n{\mathbf{Z}}$ , hence $F_{\alpha}$ is compatible with $d$ . ∎