ScalingStacks

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

• •

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