ScalingStacks

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