ScalingStacks

Proof. Let $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}^{f}}(I,J)$. There is $\tau\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}^{f}}(J,I)$ with $\ell(\tau)=0$. We have $d(\tau\circ\sigma)=\tau\circ d(\sigma)$. The isomorphism $\hat{H}_{n}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits_{{\mathbf{F}}_{2}[{\mathcal{H}}_{n}]}({\mathbf{Z}}/n)$ given by Proposition 6.2.11 restricts to an isomorphism of differential graded algebras $H_{n}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits_{{\mathbf{F}}_{2}[{\mathcal{H}}_{n}^{f}]}({\mathbf{Z}}/n)$. It follows that $d(\tau\circ\sigma)\in{\mathbf{F}}_{2}[{\mathcal{H}}_{n}^{f}]$, hence $d(\sigma)\in{\mathbf{F}}_{2}[{\mathcal{H}}_{n}^{f}]$. So, ${\mathbf{F}}_{2}[{\mathcal{H}}_{n}^{f}]$ is a differential subcategory of ${\mathbf{F}}_{2}[{\mathcal{H}}_{n}]$.

One shows similarly that ${\mathbf{F}}_{2}[{\mathcal{H}}_{n}^{+}]$ is a differential subcategory of ${\mathbf{F}}_{2}[{\mathcal{H}}_{n}]$.

Let $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}^{++}}(I,J)$. Let $(i_{1},i_{2})\in D(\sigma)$ and let $\sigma^{\prime}=\sigma^{i_{1},i_{2}}$. Given $i\in\tilde{I}$, we have $\sigma^{\prime}(i)=\sigma(i)$ if $i\not\in(i_{1}+n{\mathbf{Z}})\cup(i_{2}+n{\mathbf{Z}})$, while

$$\sigma^{\prime}(i_{1})=\sigma(i_{2})\geq i_{2}>i_{1}\text{ and }\sigma^{\prime}(i_{2})=\sigma(i_{1})>\sigma(i_{2})\geq i_{2}.$$

It follows that $\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{H}}_{n}^{++}}(I,J)$, hence $d(\sigma)\in{\mathbf{F}}_{2}[{\mathcal{H}}_{n}^{++}]$. ∎