ScalingStacks

Let $A$ be a differential algebra. We denote by $A\operatorname{\!-diff}\nolimits$ the category of (left) differential $A$-modules. Note that $\operatorname{Hom}\nolimits_{A\operatorname{\!-diff}\nolimits}(M,M^{\prime})$ is the differential $k$-module of $A$-linear maps $M\to M^{\prime}$. This is an idempotent-complete strongly pretriangulated differential category. We say that a differential $A$-module is strictly perfect if it is in $(\bar{A})^{i}$, where $A$ denotes the full subcategory of $A\operatorname{\!-diff}\nolimits$ with a unique object $A$.

A differential category ${\mathcal{C}}$ with one object $c$ is the same as the data of a differential algebra $A=\operatorname{End}\nolimits_{\mathcal{C}}(c)$. When ${\mathcal{C}}$ has a unique object $c$ and $A=\operatorname{End}\nolimits_{\mathcal{C}}(c)$, then there is an isomorphism $A\operatorname{\!-diff}\nolimits\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{C}}\operatorname{\!-diff}\nolimits,\ M\mapsto(c\mapsto M)$.

More generally, a differential category ${\mathcal{C}}$ can be viewed as a “differential algebra with several objects”. More precisely, there is an equivalence from the category of differential categories ${\mathcal{C}}$ with finitely many objects (arrows are differential functors) to the category of differential algebras $A$ equipped with a finite set $I$ of orthogonal idempotents with sum $1$ (arrows $(A,I)\to(A^{\prime},I^{\prime})$ are non-unital morphisms of differential algebras $f:A\to A^{\prime}$ such that $f(I)\subset I^{\prime}$):

• •

  to ${\mathcal{C}}$, we associate $A=\bigoplus_{c,c^{\prime}\in{\mathcal{C}}}\operatorname{Hom}\nolimits_{\mathcal{C}}(c,c^{\prime})$ and $I$ the set of projectors on objects of ${\mathcal{C}}$;

• •

  to $(A,I)$, we associate the differential category ${\mathcal{C}}$ with set of objects $I$ and $\operatorname{Hom}\nolimits_{\mathcal{C}}(e,f)=fAe$.