ScalingStacks

Let $k$ be a field of characteristic $2$. We write $\otimes$ for $\otimes_{k}$.

A differential module is a $k$-vector space $M$ endowed with an endomorphism $d$ satisfying $d^{2}=0$. We put $Z(M)=\ker d$. An element $m$ of $M$ is said to be closed when $d(m)=0$. We define $\operatorname{Hom}\nolimits$-spaces in the category $k\operatorname{\!-diff}\nolimits$ of differential modules by $\operatorname{Hom}\nolimits_{k\operatorname{\!-diff}\nolimits}(M,M^{\prime})=\operatorname{Hom}\nolimits_{k\operatorname{\!-Mod}\nolimits}(M,M^{\prime})$. That $k$-module has a differential given by $\operatorname{Hom}\nolimits(d_{M},M^{\prime})+\operatorname{Hom}\nolimits(M,d_{M^{\prime}})$. We define the category $Z(k\operatorname{\!-diff}\nolimits)$ as the subcategory of $k\operatorname{\!-diff}\nolimits$ with same objects as $k\operatorname{\!-diff}\nolimits$ and $\operatorname{Hom}\nolimits_{Z(k\operatorname{\!-diff}\nolimits)}(M,M^{\prime})=Z(\operatorname{Hom}\nolimits_{k\operatorname{\!-diff}\nolimits}(M,M^{\prime}))$.

The tensor product of vector spaces and the permutation of factors equip $k\operatorname{\!-diff}\nolimits$ and $Z(k\operatorname{\!-diff}\nolimits)$ with a structure of symmetric monoidal category.

A differential category is a category enriched over $Z(k\operatorname{\!-diff}\nolimits)$.

Let ${\mathcal{V}}$ and ${\mathcal{V}}^{\prime}$ be two differential categories. We denote by $\operatorname{Hom}\nolimits({\mathcal{V}},{\mathcal{V}}^{\prime})$ the differential category of ($k$-linear) differential functors ${\mathcal{V}}\to{\mathcal{V}}^{\prime}$. Its $\operatorname{Hom}\nolimits$ spaces are $k$-linear natural transformations.

We denote by ${\mathcal{V}}\otimes{\mathcal{V}}^{\prime}$ the differential category with set of objects $\mathrm{Obj}({\mathcal{V}})\times\mathrm{Obj}({\mathcal{V}}^{\prime})$ and with $\operatorname{Hom}\nolimits_{{\mathcal{V}}\otimes{\mathcal{V}}^{\prime}}((v_{1},v^{\prime}_{1}),(v_{2},v^{\prime}_{2}))=\operatorname{Hom}\nolimits_{{\mathcal{V}}}(v_{1},v_{2})\otimes\operatorname{Hom}\nolimits_{{\mathcal{V}}^{\prime}}(v^{\prime}_{1},v^{\prime}_{2})$.

We denote by ${\mathcal{V}}\operatorname{\!-diff}\nolimits=\operatorname{Hom}\nolimits({\mathcal{V}},k\operatorname{\!-diff}\nolimits)$ the category of ${\mathcal{V}}$-modules. There is a fully faithful embedding $v\mapsto\operatorname{Hom}\nolimits_{\mathcal{V}}(-,v):{\mathcal{V}}\to{\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits$ and we identify ${\mathcal{V}}$ with its image.

Note that $\mathrm{add}({\mathcal{V}})$ identifies with the smallest full subcategory of ${\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits$ containing ${\mathcal{V}}$ and closed under finite direct sums and isomorphisms.

There is a differential functor $\otimes_{\mathcal{V}}:{\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits\otimes{\mathcal{V}}\operatorname{\!-diff}\nolimits\to k\operatorname{\!-diff}\nolimits$. Given $M\in{\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits$ and $N\in{\mathcal{V}}\operatorname{\!-diff}\nolimits$, there is an exact sequence of differential $k$-modules

Given $v\in{\mathcal{V}}$, we have $\operatorname{Hom}\nolimits(-,v)\otimes_{\mathcal{V}}N=N(v)$ and $M\otimes_{{\mathcal{V}}}\operatorname{Hom}\nolimits(v,-)=M(v)$.

Recall that a category is idempotent complete if all idempotent maps have images.

We denote by ${\mathcal{V}}^{i}$ the idempotent completion of ${\mathcal{V}}$: this is the smallest full subcategory of ${\mathcal{V}}^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits$ containing ${\mathcal{V}}$ and closed under direct summands and isomorphisms. The $2$-functor ${\mathcal{V}}\mapsto{\mathcal{V}}^{i}$ is left adjoint to the embedding of idempotent-complete differential categories in differential categories.