ScalingStacks

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.