ScalingStacks

For a graded version, assume that $\mathcal{C}$ is equipped with an autoequivalence $\langle 1\rangle.$ Then one can define the $\mathbb{Z}$-graded locally unital algebra

(2.3)

$\displaystyle\operatorname{End}_{\mathcal{C}}^{\bullet}(\mathcal{T})=\bigoplus_{M,N\in\mathcal{T}}\operatorname{Hom}^{\bullet}_{\mathcal{C}}(M,N)\text{ where }\operatorname{Hom}^{n}_{\mathcal{C}}(M,N)=\operatorname{Hom}_{\mathcal{C}}(M,N\langle n\rangle).$

Again, general nonsense yields