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 $(M_{1}\otimes M_{2})_{g}=\bigoplus_{(g_{1},g_{2})\in p^{-1}(g)}(M_{1})_{g_{1}}\otimes(M_{2})_{g_{2}}$.

Let $\mathrm{Alg}$ be the $2$-category with objects the differential algebras, and $\operatorname{Hom}\nolimits_{\mathrm{Alg}}(A,A^{\prime})$ the category of $(A^{\prime},A)$-bimodules. The composition of $1$-arrows is the tensor product of differential bimodules.

Given $M$ an $(A^{\prime},A)$-bimodule, we put $M^{\vee}=\operatorname{Hom}\nolimits_{A^{\operatorname{opp}\nolimits}}(M,A)$, an $(A,A^{\prime})$-bimodule.

There is a morphism of $(A^{\prime},A)$-bimodules

It is an isomorphism if $M$ is finitely generated and projective as a (non-differential) $A^{\operatorname{opp}\nolimits}$-module.

There is a morphism of functors

It is an isomorphism if $M^{\vee}$ is finitely generated and projective as a (non-differential) $A$-module.

Combining those two morphisms, we obtain a morphism of functors

that is an isomorphism if $M$ is finitely generated and projective as a (non-differential) $A^{\operatorname{opp}\nolimits}$-module. So, when this holds, we have an adjoint pair $(M^{\vee}\otimes_{A^{\prime}}-,M\otimes_{A}-)$, with corresponding unit $\eta:A^{\prime}\to M\otimes_{A}M^{\vee}$ and counit $\varepsilon:M^{\vee}\otimes_{A^{\prime}}M\to A$. In other terms, the bimodule $M^{\vee}$ is a left dual of $M$.

Note conversely that given $M$ such that $(M^{\vee}\otimes_{A^{\prime}}-,M\otimes_{A}-)$ is an adjoint pair, then $M^{\vee}$ is a finitely generated projective $A$-module because $\operatorname{Hom}\nolimits_{A}(M^{\vee},-)$ is exact and commutes with direct sums, hence $M\simeq\operatorname{Hom}\nolimits_{A}(M^{\vee},A)$ is finitely generated and projective as an $A^{\operatorname{opp}\nolimits}$-module.

We say that $M$ is right finite when it is finitely generated and projective as an $A^{\operatorname{opp}\nolimits}$-module. We say that $M$ is left finite when it is finitely generated and projective as an $A^{\prime}$-module.

Consider the $2$-full subcategory $\mathrm{Alg}^{r}$ (resp. $\mathrm{Alg}^{l}$) of $\mathrm{Alg}$ with same objects and $1$-arrows the right (resp. left) finite bimodules. There is an equivalence of $2$-categories $\mathrm{Alg}^{r}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(\mathrm{Alg}^{l})^{\mathrm{rev}{\operatorname{opp}\nolimits}}$. It is the identity on objects and sends a bimodule $M$ to $M^{\vee}$.

Let ${\mathcal{C}}$ and ${\mathcal{C}}^{\prime}$ be differential categories. A $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule is a differential functor ${\mathcal{C}}\otimes{\mathcal{C}}^{\prime{\operatorname{opp}\nolimits}}\to k\operatorname{\!-diff}\nolimits$. There is a $2$-category $\mathrm{Bimod}$ of differential categories and bimodules. Its objects are differential categories and ${{\mathcal{H}}om}_{\mathrm{Bimod}}({\mathcal{C}},{\mathcal{C}}^{\prime})$ is the differential category of $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodules. Composition is given by tensor product: given ${\mathcal{C}}^{\prime\prime}$ a differential category, $M$ a $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule and $N$ a $({\mathcal{C}}^{\prime},{\mathcal{C}}^{\prime\prime})$-bimodule, we put

There is an equivalence of $2$-categories $\mathrm{Bimod}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\mathrm{Bimod}^{\mathrm{rev}}$ sending a differential category ${\mathcal{C}}$ to ${\mathcal{C}}^{\operatorname{opp}\nolimits}$ and a $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule to the same functor, viewed as a $({\mathcal{C}}^{\prime{\operatorname{opp}\nolimits}},{\mathcal{C}}^{\operatorname{opp}\nolimits})$-bimodule.

The bimodule $\operatorname{Hom}\nolimits:{\mathcal{C}}\otimes{\mathcal{C}}^{\operatorname{opp}\nolimits}\to k\operatorname{\!-diff}\nolimits,\ (c_{1},c_{2})\mapsto\operatorname{Hom}\nolimits_{\mathcal{C}}(c_{2},c_{1})$ is an identity for the tensor product. The canonical isomorphism of $({\mathcal{C}},{\mathcal{C}})$-bimodules $\operatorname{Hom}\nolimits\otimes_{\mathcal{C}}\operatorname{Hom}\nolimits\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits$ is given by

Let $M$ be a $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodule. We define the $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule $M^{\vee}$ by

There is a morphism of $({\mathcal{C}},{\mathcal{C}})$-bimodules $\varepsilon_{M}:M^{\vee}\otimes_{{\mathcal{C}}^{\prime}}M\to\operatorname{Hom}\nolimits$ given by

Given $L\in{\mathcal{C}}\operatorname{\!-diff}\nolimits$ and $L^{\prime}\in{\mathcal{C}}^{\prime}\operatorname{\!-diff}\nolimits$, we have a morphism functorial in $L$ and $L^{\prime}$

We say that $M$ is right finite if the morphism above is an isomorphism for all $L$ and $L^{\prime}$. When this holds, the functor $M^{\vee}\otimes_{{\mathcal{C}}^{\prime}}-$ is left adjoint to $M\otimes_{{\mathcal{C}}}-$ and $M^{\vee}$ is left dual to $M$ . We also write ${{}^{\vee}N}=M$ where $N=M^{\vee}$. We say that $M$ is left finite if it is a right finite $({\mathcal{C}}^{\prime{\operatorname{opp}\nolimits}},{\mathcal{C}}^{\operatorname{opp}\nolimits})$-bimodule.

Let $M$ be a $({\mathcal{C}},{\mathcal{C}})$-bimodule. We define the differential category $T_{{\mathcal{C}}}(M)$. Its objects are those of ${\mathcal{C}}$ and

There is a $2$-functor from $\mathrm{Alg}$ to $\mathrm{Bimod}$: it sends $A$ to the differential category ${\mathcal{C}}_{A}$ with one object $c_{A}$ and $\operatorname{End}\nolimits(c_{A})=A$. It sends an $(A^{\prime},A)$-bimodule $M$ to the $({\mathcal{C}}_{A^{\prime}},{\mathcal{C}}_{A})$-bimodule ${\mathcal{C}}_{M}$ given by ${\mathcal{C}}_{M}(c_{A},c_{A^{\prime}})=M$. This $2$-functor provides isomorphisms of categories $\operatorname{Hom}\nolimits_{\mathrm{Alg}}(A,A^{\prime})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{\mathrm{Bimod}}({\mathcal{C}}_{A},{\mathcal{C}}_{A^{\prime}})$.

There is a $2$-fully faithful $2$-functor from the $2$-category of differential categories to $\mathrm{Bimod}^{\mathrm{rev}}$: it sends ${\mathcal{C}}$ to ${\mathcal{C}}$ and $F:{\mathcal{C}}\to{\mathcal{C}}^{\prime}$ to the $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule $(c,c^{\prime})\mapsto\operatorname{Hom}\nolimits(c^{\prime},F(c))$.

There is a $2$-fully faithful $2$-functor from $\mathrm{Bimod}$ to the $2$-category of differential categories: it sends ${\mathcal{C}}$ to ${\mathcal{C}}\operatorname{\!-diff}\nolimits$ and $M$ a $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodule to $M\otimes_{{\mathcal{C}}}-:{\mathcal{C}}\operatorname{\!-diff}\nolimits\to{\mathcal{C}}^{\prime}\operatorname{\!-diff}\nolimits$.

Composing the $2$-functor $\mathrm{Alg}\to\mathrm{Bimod}$ and the $2$-functor from $\mathrm{Bimod}$ to the $2$-category of differential categories, we obtain a differential $2$-functor from $\mathrm{Alg}$ to the $2$-category of differential categories: it sends $A$ to $A\operatorname{\!-diff}\nolimits$ and it sends an $(A^{\prime},A)$-bimodule $M$ to the functor $M\otimes_{A}-:A\operatorname{\!-diff}\nolimits\to A^{\prime}\operatorname{\!-diff}\nolimits$. Note that this $2$-functor is $2$-fully faithful.

A pointed set is a set with a distinguished element $0$. The category $\operatorname{Sets}\nolimits^{\bullet}$ of pointed sets has objects pointed sets and arrows those maps that preserve the distinguished element.

It has coproducts: $\bigvee S_{i}$ is the quotient of $\coprod S_{i}$ by the relation identifying the $0$-objects of the $S_{i}$’s.

We define $\bigwedge S_{i}$ as the quotient of $\prod S_{i}$ by the relation identifying an element with $(0)_{i}$ if one of its components is $0$. There is a canonical isomorphism $S\wedge\{0,\ast\}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}S$. This provides the category of pointed sets with a structure of symmetric monoidal category (the tensor product of $S_{1}$ and $S_{2}$ is $S_{1}\wedge S_{2}$) and there is a symmetric monoidal functor from the category of sets to the category of pointed sets $E\mapsto E_{+}=E\sqcup\{0\}$.

Given $S$ a pointed set and $k$ a commutative ring, we denote by $k[S]$ the quotient of the free $k$-module with basis $S$ by the $k$-submodule generated by the distinguished element of $S$. This gives a coproduct preserving monoidal functor from the category of pointed sets to the category of $k$-modules.

Assume $k$ is finite. Let $S$ and $S^{\prime}$ be two pointed sets. We say that a $k$-linear map $f:k[S]\to k[S^{\prime}]$ is bounded if there is $N>0$ such that for all $s\in S$, the set of elements of $S^{\prime}$ that have a non-zero coefficient in $f(s)$ has fewer than $N$ elements.

The functor $k[-]$ induces a bijection from $k[\operatorname{Hom}\nolimits_{\operatorname{Sets}\nolimits^{\bullet}}(S,S^{\prime})]$ to the subspace of bounded maps in $\operatorname{Hom}\nolimits_{k\operatorname{\!-Mod}\nolimits}(k[S],k[S^{\prime}])$.

Let $G$ be a set. A $G$-graded pointed set is a pointed set $S$ together with pointed subsets $S_{g}$ for $g\in G$ such that $S=\bigcup_{g\in G}S_{g}$ and $S_{g}\cap S_{h}=\{0\}$ for $g\neq h$.

Given a map $f:G\to G^{\prime}$ and $S$ a $G$-graded pointed set, we define a structure of $G^{\prime}$-graded pointed set on $S$ by setting $S_{g^{\prime}}=\{0\}\cup\bigcup_{g\in f^{-1}(g^{\prime})}S_{g}$.

Given $G_{1}$ and $G_{2}$ two sets and $S_{i}$ a $G_{i}$-graded pointed set for $i\in\{1,2\}$, then $S_{1}\wedge S_{2}$ is a $(G_{1}\times G_{2})$-graded pointed set with $(S_{1}\wedge S_{2})_{(g_{1},g_{2})}=(S_{1})_{g_{1}}\wedge(S_{2})_{g_{2}}$.

Assume $G$ is a monoid. Given two $G$-graded pointed sets $S$ and $T$, there is a structure of $(G\times G)$-graded pointed set on $S\wedge T$. Via the multiplication map, we obtain a structure of $G$-graded pointed set on $S\wedge T$. This makes the category of $G$-graded pointed sets into a monoidal category with unit object the pointed set $S=\{0,\ast\}$ with $S_{1}=S$ and $S_{g}=\{0\}$ for $g\neq 1$.

Let $G$ be a poset. A $G$-filtered set (resp. pointed set) is a set (resp. a pointed set) $S$ together with subsets (resp. pointed subsets) $S_{\geq g}$ for $g\in G$ such that $S_{\geq g}\subset S_{\geq g^{\prime}}$ if $g>g^{\prime}$ and such that given $s\in S$ (resp. $s\in S\setminus\{0\}$), the set $\{g\in G\ |\ s\in S_{\geq g}\}$ is non-empty and has a maximal element, which we denote by $\deg(s)$.

Note that a structure of $G$-filtered set on a set (resp. a pointed set) $S$ is the same as the data of a map $S\to G$ (resp. a map $S\setminus\{0\}\to G$).

The associated $G$-graded pointed set is ${\operatorname{gr}\nolimits}S=\{0\}\sqcup S$ (resp. ${\operatorname{gr}\nolimits}S=S$) with

If $G$ is a (partially) ordered monoid, then the category of $G$-filtered sets (resp. pointed sets) is a monoidal category with $(S\wedge T)_{\geq g}$ the image of $\coprod_{g_{1},g_{2}\in G,g_{1}g_{2}\geq g}(S_{\geq g_{1}}\times T_{\geq g_{2}})$ in $S\wedge T$. Its unit object is the set $S=\{\ast\}$ (resp. the pointed set $S=\{0,\ast\}$) with $S_{\geq g}=S$ if $1\geq g$ and $S_{\geq g}=\emptyset$ (resp. $S_{\geq g}=\{0\}$) otherwise.

There is a monoidal functor $S\mapsto{\operatorname{gr}\nolimits}S$ from the monoidal category of $G$-filtered sets (resp. pointed sets) to the monoidal category of $G$-graded pointed sets. Given $f:S\to T$ a map between $G$-filtered sets (resp. pointed sets), the map ${\operatorname{gr}\nolimits}f:{\operatorname{gr}\nolimits}S\to{\operatorname{gr}\nolimits}T$ is given for $s\in({\operatorname{gr}\nolimits}S)_{g}$ by $({\operatorname{gr}\nolimits}f)(s)=f(s)$ if $f(s)\in({\operatorname{gr}\nolimits}T)_{g}$ and $({\operatorname{gr}\nolimits}f)(s)=0$ otherwise.

Note also that given a commutative ring $k$ there is a monoidal functor $S\mapsto k[S]$ from the category of $G$-graded pointed sets to the category of $G$-graded $k$-modules.

A pointed category is a category enriched in pointed sets. We define similarly $G$-graded pointed categories, etc. The monoidal functors ${\mathcal{V}}_{1}\to{\mathcal{V}}_{2}$ defined above provide a construction from a category enriched in ${\mathcal{V}}_{1}$ of a category enriched in ${\mathcal{V}}_{2}$. Let us describe this more explicitly.

$\bullet\$Given a $G$-filtered category (or a $G$-filtered pointed category) ${\mathcal{C}}$, we have a $G$-graded pointed category ${\operatorname{gr}\nolimits}{\mathcal{C}}$. Its objects are the same as those of ${\mathcal{C}}$ and $\operatorname{Hom}\nolimits_{{\operatorname{gr}\nolimits}{\mathcal{C}}}(c,c^{\prime})={\operatorname{gr}\nolimits}\operatorname{Hom}\nolimits_{{\mathcal{C}}}(c,c^{\prime})$.

$\bullet\$Given a pointed category ${\mathcal{C}}$, we denote by $k[{\mathcal{C}}]$ the associated $k$-linear category: its objects are those of ${\mathcal{C}}$ and $\operatorname{Hom}\nolimits_{k[{\mathcal{C}}]}(c,c^{\prime})=k[\operatorname{Hom}\nolimits_{\mathcal{C}}(c,c^{\prime})]$. If ${\mathcal{C}}$ is a $G$-graded pointed category, then $k[{\mathcal{C}}]$ is a $k$-linear $G$-graded category.

$\bullet\$Given a category ${\mathcal{C}}$, the associated pointed category ${\mathcal{C}}_{+}$ has the same objects as ${\mathcal{C}}$ and $\operatorname{Hom}\nolimits_{{\mathcal{C}}_{+}}(c,c^{\prime})=\operatorname{Hom}\nolimits_{{\mathcal{C}}}(c,c^{\prime})\sqcup\{0\}$.

Consider a family $\{{\mathcal{C}}_{i}\}$ of pointed categories. We have a pointed category $\bigwedge{\mathcal{C}}_{i}$ with object set $\prod\mathrm{Obj}({\mathcal{C}}_{i})$ and $\operatorname{Hom}\nolimits_{\bigwedge{\mathcal{C}}_{i}}((c_{i}),(c^{\prime}_{i}))=\bigwedge\operatorname{Hom}\nolimits_{{\mathcal{C}}_{i}}(c_{i},c^{\prime}_{i})$. Similarly, we have a pointed category $\bigvee{\mathcal{C}}_{i}$ with object set $\coprod\mathrm{Obj}({\mathcal{C}}_{i})$ and given $c\in{\mathcal{C}}_{r}$ and $c^{\prime}\in{\mathcal{C}}_{s}$, we have

Note that the data of a structure of $G$-filtered pointed category on a pointed category ${\mathcal{C}}$ is the same as the data of a map $\deg$ from the set of non-zero maps of ${\mathcal{C}}$ to $G$ such that $\deg(\beta\circ\alpha)\geq\deg(\beta)\deg(\alpha)$ for any two composable maps $\alpha$ and $\beta$ such that $\beta\circ\alpha\neq 0$.

Given a $G$-filtered pointed category ${\mathcal{C}}$ with degree function $\deg$ and given a morphism of (partially) ordered monoids $f:G\to H$, we obtain a structure of $H$-filtered pointed category on ${\mathcal{C}}$ with degree function $f\circ\deg$.

Note that the category $\operatorname{Sets}\nolimits^{\bullet}$ has a structure of pointed category: the distinguished map between two pointed sets is the map with image $0$.

We define a differential pointed set to be a pointed set $S$ together with a bounded endomorphism $d$ of ${\mathbf{F}}_{2}[S]$ satisfying $d^{2}=0$.

Given $S$ and $S^{\prime}$ two differential pointed sets, then $S\vee S^{\prime}$ and $S\wedge S^{\prime}$ have structures of differential pointed sets coming from the canonical isomorphisms ${\mathbf{F}}_{2}[S\vee S^{\prime}]\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{F}}_{2}[S]\oplus{\mathbf{F}}_{2}[S^{\prime}]$ and ${\mathbf{F}}_{2}[S\wedge S^{\prime}]\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{F}}_{2}[S]\otimes{\mathbf{F}}_{2}[S^{\prime}]$.

We define the category $\mathrm{diff}$ of differential pointed sets: its objects are differential pointed sets and maps the maps of pointed sets. There is a functor ${\mathbf{F}}_{2}[-]:\mathrm{diff}\to{\mathbf{F}}_{2}\operatorname{\!-diff}\nolimits$. Let $S$ and $S^{\prime}$ be two differential pointed sets. Because the differentials on ${\mathbf{F}}_{2}[S]$ and ${\mathbf{F}}_{2}[S^{\prime}]$ are bounded, the vector space ${\mathbf{F}}_{2}[\operatorname{Hom}\nolimits_{\operatorname{Sets}\nolimits^{\bullet}}(S,S^{\prime})]$ identifies with a subspace of $\operatorname{Hom}\nolimits_{{\mathbf{F}}_{2}\operatorname{\!-Mod}\nolimits}({\mathbf{F}}_{2}[S],{\mathbf{F}}_{2}[S^{\prime}])$ that is stable under the differential $\operatorname{Hom}\nolimits(d_{{\mathbf{F}}_{2}[S]},-)+\operatorname{Hom}\nolimits(-,d_{{\mathbf{F}}_{2}[S^{\prime}]})$.

We define $Z(\mathrm{diff})$ as the subcategory of $\mathrm{diff}$ with same objects as $\mathrm{diff}$ and with $\operatorname{Hom}\nolimits_{Z(\mathrm{diff})}(S,S^{\prime})$ the subset of maps in the kernel of $d$ (where we view $\operatorname{Hom}\nolimits_{\mathrm{diff}}(S,S^{\prime})$ inside $\operatorname{Hom}\nolimits_{{\mathbf{F}}_{2}\operatorname{\!-Mod}\nolimits}({\mathbf{F}}_{2}[S],{\mathbf{F}}_{2}[S^{\prime}])$). The categories $\mathrm{diff}$ and $Z(\mathrm{diff})$ have a structure of symmetric monoidal category coming from those on pointed sets and differential modules.

We define a differential pointed category to be a category enriched in $Z(\mathrm{diff})$. This is the same as a pointed category ${\mathcal{V}}$ together with a differential on ${\mathbf{F}}_{2}[{\mathcal{V}}]$ endowing it with a structure of differential category. The $2$-functor ${\mathcal{V}}\mapsto{\mathbf{F}}_{2}[{\mathcal{V}}]$ from the $2$-category of differential pointed categories to the $2$-category of differential categories is $2$-faithful and $2$-conservative.

Note that the category $\mathrm{diff}$ is a differential pointed category:

All our constructions below for differential pointed categories are compatible with the corresponding constructions for differential categories, via the $2$-functor ${\mathbf{F}}_{2}[?]$.

Given $G$ a ${\mathbf{Z}}$-monoid, we will also consider differential $G$-graded pointed sets: these are differential pointed sets $S$ with a structure of $G$-graded pointed set such that $d(S_{g})]\subset{\mathbf{F}}_{2}[S_{g+1}]$ for $g\in G$. We have a corresponding notion of differential $G$-graded pointed category.

Let ${\mathcal{V}}$ be a differential pointed category. We say that a map of ${\mathcal{V}}$ is closed if its image in ${\mathbf{F}}_{2}[{\mathcal{V}}]$ is closed. Given $f:S\to S^{\prime}$ a closed map of differential pointed sets, we define the cone $\operatorname{cone}\nolimits(f)$ of $f$ as the pointed set $S\vee S^{\prime}$ with differential on ${\mathbf{F}}_{2}[S\vee S^{\prime}]={\mathbf{F}}_{2}[S]\oplus{\mathbf{F}}_{2}[S^{\prime}]$ given by $\left(\begin{matrix}d_{{\mathbf{F}}_{2}[S]}&0\\ f&d_{{\mathbf{F}}_{2}[S^{\prime}]}\end{matrix}\right)$.

We define a ${\mathcal{V}}$-module to be a differential pointed functor (i.e., a functor enriched in $Z(\mathrm{diff})$) ${\mathcal{V}}\to\mathrm{diff}$. We denote by ${\mathcal{V}}\operatorname{\!-diff}\nolimits$ the category of ${\mathcal{V}}$-modules.

Given $f:v_{1}\to v_{2}$ a closed map in ${\mathcal{V}}$, we define $\operatorname{cone}\nolimits(f)=\operatorname{cone}\nolimits(\operatorname{Hom}\nolimits_{{\mathcal{V}}}(f,-))\in{\mathcal{V}}\operatorname{\!-diff}\nolimits$.

Let $M$ be a ${\mathcal{V}}^{\operatorname{opp}\nolimits}$-module and $N$ a ${\mathcal{V}}$-module. We define the differential pointed set $M\wedge_{\mathcal{V}}N$ as the coequalizer of

Given ${\mathcal{V}}^{\prime}$ a differential pointed category, we define a $({\mathcal{V}},{\mathcal{V}}^{\prime})$-bimodule to be a differential pointed functor ${\mathcal{V}}\bigwedge{\mathcal{V}}^{\prime{\operatorname{opp}\nolimits}}\to\mathrm{diff}$.

Given ${\mathcal{V}}^{\prime\prime}$ a differential pointed category, $N$ a $({\mathcal{V}},{\mathcal{V}}^{\prime})$-bimodule and $M$ a $({\mathcal{V}}^{\prime},{\mathcal{V}}^{\prime\prime})$-bimodule, then $N\wedge_{{\mathcal{V}}^{\prime}}M$ is a $({\mathcal{V}},{\mathcal{V}}^{\prime\prime})$-bimodule. This gives rise to a $2$-category $\mathrm{Bimod}^{\bullet}$ of differential pointed categories and bimodules, with a $2$-fully faithful functor to the $2$-category of differential pointed categories and a $2$-faithful functor ${\mathbf{F}}_{2}[-]$ to the $2$-category $\mathrm{Bimod}$.

Let $M$ be a $({\mathcal{V}},{\mathcal{V}})$-bimodule. We define a differential pointed category $T_{{\mathcal{V}}}(M)$. Its objects are those of ${\mathcal{V}}$ and

Let us reformulate the definitions of the previous sections in terms of ${\mathbf{F}}_{2}$-vector spaces with a basis.

The functor ${\mathbf{F}}_{2}[-]$ gives an equivalence from the category of pointed sets to the category with objects ${\mathbf{F}}_{2}$-vector spaces with a basis and where maps are ${\mathbf{F}}_{2}$-linear maps sending a basis element to a basis element or $0$.

Under this equivalence, we have the following correspondences:

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  a coproduct of pointed spaces corresponds to a direct sum with basis the union of bases

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  a wedge product of pointed spaces corresponds to a tensor product with basis the product of bases

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  a $G$-graded pointed set corresponds to a $G$-graded ${\mathbf{F}}_{2}$-vector space with a basis consisting of homogeneous elements

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  a $G$-filtered pointed set corresponds to a $G$-filtered ${\mathbf{F}}_{2}$-vector space $V$, ie a family $\{V_{\geq g}\}_{g\in G}$ of subspaces of $V$ with $V_{\geq g}\subset V_{\geq g^{\prime}}$ if $g>g^{\prime}$, with a basis $B$ such that $B\cap V_{\geq g}$ is a basis of $V_{\geq g}$ for all $g\in G$ and such that given $v\in V\setminus\{0\}$, the set $\{g\in G\ |\ V_{\geq g}\neq 0\}$ is non-empty and has a maximal element

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  a differential pointed set corresponds to an ${\mathbf{F}}_{2}$-vector space with a basis together with a bounded differential.