ScalingStacks

Let ${\mathcal{C}}$ be a category. We denote by ${\mathcal{C}}^{\operatorname{opp}\nolimits}$ the opposite category. We identify ${\mathcal{C}}$ with a full subcategory of $\operatorname{Hom}\nolimits({\mathcal{C}}^{\operatorname{opp}\nolimits},\mathrm{Sets})$ via the Yoneda embedding $c\mapsto\operatorname{Hom}\nolimits(-,c)$.

Given $(L,R)$ a pair of adjoint functors, we denote the unit of the adjunction by $\eta_{L,R}$ and the counit by $\varepsilon_{L,R}$.

When ${\mathcal{C}}$ is enriched in abelian groups, we denote by $\operatorname{add}\nolimits({\mathcal{C}})$ the smallest full subcategory of $\operatorname{Hom}\nolimits({\mathcal{C}}^{\operatorname{opp}\nolimits},\mathrm{Sets})$ containing ${\mathcal{C}}$ and closed under finite coproducts and isomorphisms.

Let ${\mathcal{X}}$ be a $2$-category. We denote by ${\mathcal{X}}^{\operatorname{opp}\nolimits}$ the $2$-category with same objects and ${{\mathcal{H}}om}(x,y)={{\mathcal{H}}om}(x,y)^{\operatorname{opp}\nolimits}$. We denote by ${\mathcal{X}}^{\mathrm{rev}}$ the $2$-category with the same objects and with ${{\mathcal{H}}om}(x,y)={{\mathcal{H}}om}(y,x)$ for $x$ and $y$ two objects of ${\mathcal{X}}$ (so that the composition of $1$-arrows is reversed).

Let ${\mathcal{C}}{at}$ be the $2$-category of categories. There is an equivalence ${\mathcal{C}}{at}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{C}}{at}^{\operatorname{opp}\nolimits}$ sending a category ${\mathcal{C}}$ to ${\mathcal{C}}^{\operatorname{opp}\nolimits}$.

Let ${\mathcal{C}}{at}^{r}$ (resp. ${\mathcal{C}}{at}^{l}$) be the $2$-full $2$-subcategory of ${\mathcal{C}}{at}$ with $1$-arrows those functors that admit a left (resp. right) adjoint. There is an equivalence of $2$-categories ${\mathcal{C}}{at}^{r}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}({\mathcal{C}}{at}^{l})^{\mathrm{rev}{\operatorname{opp}\nolimits}}$. It is the identity on objects and sends a functor to a left adjoint.

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.

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.

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 $f(I)\subset I^{\prime}$):

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  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 ${\mathcal{C}}$;

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

We define a ${\mathbf{Z}}$-monoid $G$ to be a monoid $G$ endowed with an action of the group ${\mathbf{Z}}$, denoted by $g\mapsto g+n$ for $g\in G$ and $n\in{\mathbf{Z}}$, and such that $(g+n)(g^{\prime}+n^{\prime})=gg^{\prime}+n+n^{\prime}$. Note that $e_{G}+{\mathbf{Z}}$ is a central submonoid of $G$, where $e_{G}$ denotes the unit of $G$. So, the data above is equivalent to the data of a morphism of monoids ${\mathbf{Z}}\to Z(G)$. This is itself determined by the image of $1$, a central invertible element $\upsilon$ of $G$.

We define a differential $G$-graded $k$-module to be a $G$-graded $k$-module $M$ together with a differential module structure such that $d(M_{g})\subset M_{g+1}$ (cf [LiOzTh1, §2.5]).

Given $g\in G$, we define $M\langle g\rangle$ to be the differential $G$-graded $k$-module given by $(M\langle g\rangle)_{h}=M_{hg}$. Similarly, we define $\langle g\rangle M$ by $(\langle g\rangle M)_{h}=M_{gh}$.

We define similarly the notion of differential $G$-graded algebra, of differential $G$-graded category, etc.

When $G={\mathbf{Z}}$ and $\upsilon=1$, we recover the usual notion of differential graded $k$-module, etc.

Let $G_{1}$ and $G_{2}$ be two ${\mathbf{Z}}$-monoids. We define $G_{1}\times_{{\mathbf{Z}}}G_{2}$ as the quotient of $G_{1}\times G_{2}$ by the equivalence relation $(g_{1},g_{2}+n)\sim(g_{1}+n,g_{2})$ for $g_{1},g_{2}\in G$ and $n\in{\mathbf{Z}}$. Denote by $p:G_{1}\times G_{2}\to G_{1}\times_{{\mathbf{Z}}}G_{2}$ the quotient map, a morphism of monoids. There is a structure of ${\mathbf{Z}}$-monoid on $G_{1}\times_{{\mathbf{Z}}}G_{2}$ given by $p(g_{1},g_{2})+1=p(g_{1}+1,g_{2})=p(g_{1},g_{2}+1)$.

Let $M_{i}$ be a differential $G_{i}$-graded $k$-module for $i\in\{1,2\}$. We define a structure of differential $(G_{1}\times_{{\mathbf{Z}}}G_{2})$-module on the differential module $M_{1}\otimes M_{2}$ by setting $(M_{1}\otimes M_{2})_{g}=\bigoplus_{(g_{1},g_{2})\in p^{-1}(g)}(M_{1})_{g_{1}}\otimes(M_{2})_{g_{2}}$.