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

Rasool Hafezi, Hideto Asashiba, Mohammad Hossein Keshavarz

Abstract

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, $\mathcal{C}/G$ the orbit category of $\mathcal{C}$, $(P, φ) : \mathcal{C} \rightarrow \mathcal{C}/G$ the canonical $G$-covering, and $\mathrm{mod}\mbox{-} \mathcal{C}$, $\mathrm{mod}\mbox{-} (\mathcal{C}/G)$ the categories of finitely generated modules over $\mathcal{C}, \mathcal{C}/G$, respectively. Then it is well known that there exists a canonical G-precovering $(P., φ.) : \mathrm{mod}\mbox{-} \mathcal{C} \rightarrow \mathrm{mod}\mbox{-} (\mathcal{C}/G)$. By applying the machinery above to this $(P., φ.)$, new $G$-precoverings $(\mathrm{mod}\mbox{-} \mathcal{C}) / S \rightarrow (\mathrm{mod}\mbox{-} \mathcal{C}/G)/S'$ are induced between the factor categories or localizations of $\mathrm{mod}\mbox{-} \mathcal{C}$ and $\mathrm{mod}\mbox{-} \mathcal{C}/G$, respectively. This is further applied to the morphism category $\mathrm{H}(\mathrm{mod}\mbox{-} \mathcal{C})$ of $\mathrm{mod}\mbox{-} \mathcal{C}$ to have a $G$-precovering $\mathrm{fp}(\mathcal{K}) \rightarrow \mathrm{fp}(\mathcal{K}')$ between the categories of finitely presented modules over suitable subcategories $\mathcal{K}$ and $\mathcal{K}'$ of $\mathrm{mod}\mbox{-}\mathcal{C}$ and $ \mathrm{mod}\mbox{-} \mathcal{C}/G$, respectively.

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

Abstract

Throughout this paper is a fixed group, and is a fixed field. All categories are assumed to be -linear. First we give a systematic way to induce -precoverings by adjoint functors using a 2-categorical machinery, which unifies many similar constructions of -precoverings. Now let be a skeletally small category with a -action, the orbit category of , the canonical -covering, and , the categories of finitely generated modules over , respectively. Then it is well known that there exists a canonical G-precovering . By applying the machinery above to this , new -precoverings are induced between the factor categories or localizations of and , respectively. This is further applied to the morphism category of to have a -precovering between the categories of finitely presented modules over suitable subcategories and of and , respectively.
Paper Structure (21 sections, 45 theorems, 143 equations)

This paper contains 21 sections, 45 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Examples of constructions of $G$-precoverings
  3. Unification
  4. Applications
  5. Relations with other works
  6. Preliminary
  7. Orbit categories
  8. Functor categories
  9. Pushdown functor
  10. Locally bounded categories
  11. Extension of a 2-endofunctor of $k\text{-}\mathbf{CAT}$ to $G\text{-}\mathbf{CAT}$
  12. $G$-precoverings induced by adjoint
  13. Preservation of compactness
  14. $G$-precoverings induced by adjoint
  15. Main theorems
  16. ...and 6 more sections

Key Result

Theorem 1.2

Consider the following diagram of $2$-categories, $2$-functors, and a strict $2$-natural transformation on the left, and its extension by Lemma lem:ext-2-nat on the right: where $\mathsf{C}$ is a $2$-subcategory of $k\text{-}\mathbf{CAT}$ and $\sigma$ is the inclusion $2$-functor. Assume the following conditions for all $\mathcal{C} \in \mathsf{C}$: Now let $(\mathcal{A}, A)$ be a $G$-category i

Theorems & Definitions (119)

  • Definition 1.1: see Definition \ref{['dfn:precov-adj']}
  • Theorem 1.2: see Theorem \ref{['thm:factor-case']}
  • Theorem 1.3: see Proposition \ref{['Prp:factor-covering']}
  • Theorem 1.4: see Proposition \ref{['Prp:localization-covering']}
  • Theorem 1.5: see Proposition \ref{['Prop-pushdown-adj']}
  • Theorem 1.6: see Theorem \ref{['thm:G-Fp-precovering']}
  • Definition 2.1: $G$-Categories
  • Definition 2.2: $G$-Invariant Functors
  • Definition 2.3: $G$-equivariant functors
  • Definition 2.4: Morphisms between $G$-equivariant functors
  • ...and 109 more