ScalingStacks

Let ${\mathcal{C}}$ be a pointed category. We define a pointed category $S({\mathcal{C}})$. Its objects are finite families $I$ of distinct objects of ${\mathcal{C}}$. We put

where $\phi$ runs over the set of bijections $I\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}J$.

An element of $\operatorname{Hom}\nolimits_{S({\mathcal{C}})}(I,J)$ is a pair $(\phi,f)$ where $\phi:I\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}J$ is a bijection and $f\in\prod_{i\in I}\operatorname{Hom}\nolimits_{{\mathcal{C}}}(i,\phi(i))$. All pairs with $f_{i}=0$ for some $i$ are identified, and they form the $0$-element of $\operatorname{Hom}\nolimits_{S({\mathcal{C}})}(I,J)$. The composition is given by $(\psi,g)\circ(\phi,f)=(\psi\phi,(g_{\phi(i)}\circ f_{i})_{i\in I})$.

Given a functor $F:{\mathcal{C}}\to{\mathcal{C}}^{\prime}$ of pointed categories that is injective on the set of objects, we obtain a functor $S(F):S({\mathcal{C}})\to S({\mathcal{C}}^{\prime})$ of pointed categories. If in addition $F$ is faithful, then $S(F)$ is faithful.

Given a commutative ring $k$ and a $k$-linear category ${\mathcal{D}}$, we define a $k$-linear category $S_{k}({\mathcal{D}})$. Its objects are finite families $I$ of distinct objects of ${\mathcal{D}}$. We put

The composition is defined as in the case of pointed categories above.

Consider a functor $F:{\mathcal{D}}\to{\mathcal{D}}^{\prime}$ of $k$-linear categories that is injective on the set of objects. We obtain a functor $S_{k}(F):S_{k}({\mathcal{D}})\to S_{k}({\mathcal{D}}^{\prime})$ of pointed categories. If $\operatorname{Hom}\nolimits$-spaces in ${\mathcal{D}}$ and ${\mathcal{D}}^{\prime}$ are flat over $k$ and $F$ is faithful, then $S_{k}(F)$ is faithful.

Given a pointed category ${\mathcal{C}}$, there is an isomorphism of $k$-linear categories $k[S({\mathcal{C}})]\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}S_{k}(k[{\mathcal{C}}])$.