Lemma 6.2.8 (0P7S) · ScalingStacks

0P7S

Lemma 6.2.8. Let $\sigma,\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ . Given $\tau\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,I)$ with $\ell(\tau)=0$ , we have $\sigma^{\prime}<\sigma$ if and only if $\tau\circ\sigma^{\prime}<\tau\circ\sigma$ if and only if $\sigma^{\prime}\circ\tau<\sigma\circ\tau$ .

0P7T

Proof. Note that $\tau$ is an increasing bijection since $\ell(\tau)=0$ . We have $D(\tau\circ\sigma)=D(\sigma)$ and given $(i_{1},i_{2})\in D(\sigma)$ , we have $(\tau\circ\sigma)^{i_{1},i_{2}}=\tau\circ\sigma^{i_{1},i_{2}}$ . This shows the first equivalence. The second equivalence follows from the fact that $D(\sigma\circ\tau)=(\tau^{-1}\times\tau^{-1})(D(\sigma))$ and given $(i_{1},i_{2})\in D(\sigma)$ , we have $(\sigma\circ\tau)^{\tau^{-1}(i_{1}),\tau^{-1}(i_{2})}=\sigma^{i_{1},i_{2}}\circ\tau$ . ∎