ScalingStacks

Given $v_{1},v_{2}$ two objects of ${\mathcal{V}}$ and given $f\in Z\operatorname{Hom}\nolimits_{\mathcal{V}}(v_{1},v_{2})$, the cone of $f$ is the object $\mathrm{cone}(\operatorname{Hom}\nolimits_{{\mathcal{V}}}(-,f))$ of ${\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits$ denoted by . We say that ${\mathcal{V}}$ is strongly pretriangulated if the cone of any map of ${\mathcal{V}}$ is isomorphic to an object of ${\mathcal{V}}$. Note that ${\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits$ is strongly pretriangulated.

We denote by $\bar{{\mathcal{V}}}$ the smallest full strongly pretriangulated subcategory of ${\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits$ closed under taking isomorphic objects and containing ${\mathcal{V}}$. Note that $(\bar{{\mathcal{V}}})^{i}$ is strongly pretriangulated. Note also that if ${\mathcal{V}}$ is a full subcategory of a strongly pretriangulated ${\mathcal{V}}^{\prime}$, then ${\mathcal{V}}$ is strongly pretriangulated if the cone in ${\mathcal{V}}^{\prime}$ of a map between objects of ${\mathcal{V}}$ is isomorphic to an object of ${\mathcal{V}}$.

Let $v_{1},\ldots,v_{n}$ be objects of ${\mathcal{V}}$ and $f_{ij}\in\operatorname{Hom}\nolimits_{{\mathcal{V}}}(v_{j},v_{i})$ for $i<j$. Assume $d(f_{ij})=\sum_{i<r<j}f_{ir}\circ f_{rj}$ for all $i<j$. We define the twisted object $[v_{n}\oplus\cdots\oplus v_{1},\left(\begin{matrix}0\\ f_{n-1,n}&\ddots\\ \vdots&\ddots&0\\ f_{1,n}&\ldots&f_{1,2}&0\end{matrix}\right)]$ of $\bar{{\mathcal{V}}}$ inductively on $n$ as the cone of

The objects of $\bar{{\mathcal{V}}}$ are the objects of ${\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits$ isomorphic to a twisted object of ${\mathcal{V}}$.

If ${\mathcal{V}}^{\prime}$ is strongly pretriangulated, then the restriction functor $\operatorname{Hom}\nolimits(\bar{{\mathcal{V}}},{\mathcal{V}}^{\prime})\to\operatorname{Hom}\nolimits({\mathcal{V}},{\mathcal{V}}^{\prime})$ is an equivalence. So, ${\mathcal{V}}\mapsto\bar{{\mathcal{V}}}$ is left adjoint to the embedding of strongly pretriangulated differential categories in differential categories.