Subsection 4.2.2. (0PFJ) · ScalingStacks

4.2.2. Category

Let ${\mathcal{W}}$ be a differential category endowed with a lax action $(E_{i,j})$ of ${\mathcal{U}}^{2}$ .

We define a differential category $\Delta_{E}{\mathcal{W}}$ .

$\bullet\$ The objects of $\Delta_{E}{\mathcal{W}}$ are pairs $(m,\varsigma)$ where $m\in\overline{{\mathcal{W}}}^{i}$ and $\varsigma\in Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(E_{0,1}E_{1,0}(m),m)$ such that for all $i\geq 1$ , there exists $\varsigma_{i}\in Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(E_{i,i}(m),m)$ such that the composition $b_{i}$

(4.2.2)

b_{i}:(E_{0,1}E_{1,0})^{i}(m)\xrightarrow{(E_{0,1}E_{1,0})^{i-1}\varsigma}(E_{0,1}E_{1,0})^{i-1}(m)\xrightarrow{(E_{0,1}E_{1,0})^{i-2}\varsigma}\cdots\to E_{0,1}E_{1,0}(m)\xrightarrow{\varsigma}m

is equal to

(E_{0,1}E_{1,0})^{i}(m)\xrightarrow{{\mathrm{can}}}E_{i,i}(m)\xrightarrow{\varsigma_{i}}m

and $\varsigma_{i}\circ(T_{r}\otimes 1)=\varsigma_{i}\circ(1\otimes T_{r})$ for $1\leq r<i$ .

$\bullet\$ $\operatorname{Hom}\nolimits_{\Delta_{E}{\mathcal{W}}}((m,\varsigma),(m^{\prime},\varsigma^{\prime}(m))$ is the differential submodule of $\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(m,m^{\prime})$ of elements $f$ such that the following diagram commutes

The composition of maps is defined by restricting that of $\overline{{\mathcal{W}}}^{i}$ . So, we have a faithful forgetful functor $\omega:\Delta_{E}{\mathcal{W}}\to\overline{{\mathcal{W}}}^{i},\ (m,\varsigma)\mapsto m$ . Note that $\Delta_{E}{\mathcal{W}}$ is strongly pretriangulated and idempotent-complete.

0P5M

Remark 4.2.3. Note that applying the self-equivalence $(a,b)\mapsto(b,a)$ of ${\mathcal{U}}^{2}$ provides another lax action $E^{\prime}$ of ${\mathcal{U}}^{2}$ on ${\mathcal{W}}$ . The corresponding differential category $\Delta_{E^{\prime}}{\mathcal{W}}$ is not equivalent to $\Delta_{E}{\mathcal{W}}$ in general.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2