[Paper Review] Right $n$-angulated categories arising from covariantly finite subcategories

Zengqiang Lin|arXiv (Cornell University)|Sep 10, 2014

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper introduces right $n$-angulated categories as a generalization of right triangulated categories and proves that the quotient category $\mathcal{C}/\mathcal{X}$, where $\mathcal{X}$ is a covariantly finite subcategory of an additive or $n$-angulated category $\mathcal{C}$, inherits a right $n$-angulated structure under suitable conditions. The key contribution is a categorical construction that generalizes prior results on quotients of triangulated and $n$-angulated categories.

ABSTRACT

We define the notion of right $n$-angulated category, which generalizes the notion of right triangulated category. Let $\mathcal{C}$ be an additive category or $n$-angulated category and $\mathcal{X}$ a covariantly finite subcategory, we show that under certain conditions the quotient $\mathcal{C}/\mathcal{X}$ is a right $n$-angulated category. This result generalizes some previous work.

Motivation & Objective

• To generalize the notion of right triangulated categories to higher dimensions via $n$-angulated structures.
• To establish conditions under which the quotient category $\mathcal{C}/\mathcal{X}$ inherits a right $n$-angulated structure when $\mathcal{X}$ is a covariantly finite subcategory.
• To unify and extend previous results on quotient categories in higher homological algebra, including those from Frobenius $n$-exact categories and mutation pairs.
• To provide a framework for constructing new examples of right $n$-angulated categories from existing $n$-angulated or additive categories.

Proposed method

• Define a right $n$-angulated category as a triple $(\mathcal{C}, \Sigma, \Theta)$ satisfying axioms analogous to triangulated categories but for $n$-term sequences.
• Introduce the concept of $n$-cokernel and $n$-pushout to handle higher-dimensional morphism compositions in the quotient setting.
• Construct standard right $n$-angles in $\mathcal{C}/\mathcal{X}$ using left $\mathcal{X}$-approximations and $n$-angles in $\mathcal{C}$ with objects in $\mathcal{X}$.
• Verify that the class $\Theta$ of $n$-angles in $\mathcal{C}/\mathcal{X}$ satisfies the axioms (RN1)–(RN4), ensuring the structure is a right $n$-angulated category.
• Use the functor $T: \mathcal{C}/\mathcal{X} \to \mathcal{C}/\mathcal{X}$ defined by $TA = B$ for a given $n$-angle to define the shift functor in the quotient.
• Apply the construction to special cases such as $n=3$ (recovering right triangulated categories) and Frobenius $n$-angulated categories.

Experimental results

Research questions

• RQ1 Under what conditions does the quotient category $\mathcal{C}/\mathcal{X}$ inherit a right $n$-angulated structure when $\mathcal{X}$ is a covariantly finite subcategory of $\mathcal{C}$?
• RQ2 How can the notion of right $n$-angulated categories be used to generalize Happel’s Theorem and other results on quotient categories?
• RQ3 What is the role of $n$-cokernels and $n$-pushouts in constructing morphisms and completing diagrams in the quotient category?
• RQ4 In what cases does the quotient $\mathcal{C}/\mathcal{X}$ become an $n$-angulated category rather than just right $n$-angulated?
• RQ5 How do the properties of injective and projective objects in $\mathcal{C}$ affect the structure of $\mathcal{C}/\mathcal{I}$ and $\mathcal{C}/\mathcal{P}$?

Key findings

• The quotient category $\mathcal{C}/\mathcal{X}$ is a right $n$-angulated category when $\mathcal{X}$ is a covariantly finite subcategory of an additive category $\mathcal{C}$ and every $\mathcal{X}$-monic has a cokernel.
• The construction of standard right $n$-angles in $\mathcal{C}/\mathcal{X}$ is based on $n$-angles in $\mathcal{C}$ with initial morphism a left $\mathcal{X}$-approximation and intermediate terms in $\mathcal{X}$.
• For $n=3$, the result recovers the classical theorem that the quotient of a triangulated category modulo a covariantly finite subcategory is right triangulated.
• If $\mathcal{C}$ is an $n$-angulated category with enough injectives, then $\mathcal{C}/\mathcal{I}$ is a right $n$-angulated category.
• If $\mathcal{C}$ is Frobenius (has enough injectives and projectives with $\mathcal{I} = \mathcal{P}$), then $\mathcal{C}/\mathcal{I}$ is an $n$-angulated category.
• When $\mathcal{C}$ is $n$-angulated and $\mathcal{X}$ is functorially finite with $(\mathcal{C},\mathcal{C})$ an $\mathcal{X}$-mutation pair, the quotient $\mathcal{C}/\mathcal{X}$ is $n$-angulated.