ScalingStacks

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 $0$.