2-categorical approach to unifying constructions of precoverings and its applications

Throughout this paper $G$ is a fixed group, and $k$ is a fixed field. All categories are assumed to be $k$-linear. First we give a systematic way to induce $G$-precoverings by adjoint functors using a 2-categorical machinery, which unifies many similar constructions of $G$-precoverings. Now let $\mathcal{C}$ be a skeletally small category with a $G$-action, $\mathcal{C}\mathrm{/}G$ the orbit category of $\mathcal{C}$, $(P,\phi ):\mathcal{C}\to \mathcal{C}\mathrm{/}G$ the canonical $G$-covering, and $\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C}$, $\mathrm{m}\mathrm{o}\mathrm{d}-(\mathcal{C}\mathrm{/}G)$ the categories of finitely generated modules over $\mathcal{C},\mathcal{C}\mathrm{/}G$, respectively. Then it is well known that there exists a canonical G-precovering $(P\mathrm{.},\phi \mathrm{.}):\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C}\to \mathrm{m}\mathrm{o}\mathrm{d}-(\mathcal{C}\mathrm{/}G)$. By applying the machinery above to this $(P\mathrm{.},\phi \mathrm{.})$, new $G$-precoverings $(\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C})\mathrm{/}S\to (\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C}\mathrm{/}G)\mathrm{/}{S}^{\mathrm{\prime }}$ are induced between the factor categories or localizations of $\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C}$ and $\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C}\mathrm{/}G$, respectively. This is further applied to the morphism category $\mathrm{H}(\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C})$ of $\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C}$ to have a $G$-precovering $\mathrm{f}\mathrm{p}(\mathcal{K})\to \mathrm{f}\mathrm{p}({\mathcal{K}}^{\mathrm{\prime }})$ between the categories of finitely presented modules over suitable subcategories $\mathcal{K}$ and ${\mathcal{K}}^{\mathrm{\prime }}$ of $\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C}$ and $\mathrm{m}\mathrm{o}\mathrm{d}-\mathcal{C}\mathrm{/}G$, respectively.