(n+2)-Angulated Categories

• (n+2)-angulated categories are higher homological structures that generalize triangulated categories by replacing triangles with extended (n+2)-angles.
• They satisfy axioms like rotation invariance and a higher octahedral property, streamlining the construction of complex categorical mappings.
• These categories find applications in higher representation theory, cluster tilting, and the analysis of singularities in non-commutative geometry.

An $(n+2)$-angulated category is a higher homological structure generalizing the notion of triangulated categories, in which distinguished triangles are replaced by longer “$(n+2)$-angles” of objects and morphisms, and new axioms—motivated by applications in higher Auslander–Reiten theory, representation theory, and non-commutative algebraic geometry—are satisfied. These categories are central to high-dimensional homological algebra and unify the higher analogues of exactness, cluster tilting, and mutation phenomena.

1. Foundational Framework and Definition

An $(n+2)$-angulated category $(\mathcal{C}, \Sigma, \Theta)$ consists of an additive category $\mathcal{C}$, an auto-equivalence $\Sigma: \mathcal{C} \to \mathcal{C}$ (the $n$-suspension, sometimes called the shift or suspension functor), and a distinguished class $\Theta$ of $(n+2)$–$\Sigma$–sequences (called $(n+2)$-angles). A typical such sequence is

$$X^0 \xrightarrow{f^0} X^1 \xrightarrow{f^1} X^2 \to \cdots \xrightarrow{f^{n}} X^{n+1} \xrightarrow{f^{n+1}} \Sigma X^0\,.$$

The category is subject to four main axioms (see (Jasso, 2014, Arentz-Hansen et al., 2016, He et al., 2020)):

• (N1): Closure under isomorphisms, direct sums, direct summands; existence of trivial angles; and for every morphism, existence of an $(n+2)$-angle extending it.
• (N2): (Left) rotation invariance: the class of angles is preserved by left rotation.
• (N3): Morphism axiom: every commutative square of initial terms can be extended to a morphism of $(n+2)$-angles. (This axiom is in fact redundant, as it follows from (N1)(c) and (N4) (Arentz-Hansen et al., 2016).)
• (N4): Higher octahedral or mapping cone axiom: provides a higher-dimensional analogue of the triangulated category's octahedral axiom, ensuring compatibility of mapping cones between $(n+2)$-angles.

A key perspective is that, when $n=1$, one recovers the definition of a triangulated category.

2. Stable Categories of Frobenius $n$-Exact and $n$-Exangulated Categories

Let $(\mathcal{M}, \mathcal{X})$ be a Frobenius $n$-exact category in which projective and injective objects coincide. The stable category $\underline{\mathcal{M}} = \mathcal{M}/\mathcal{I}$, where $\mathcal{I}$ is the full subcategory of projective-injective objects, inherits a canonical $(n+2)$-angulated structure (Jasso, 2014). Explicitly:

• The suspension functor $\Sigma$ is given by assigning to $M$ the cokernel of a chosen $n$-exact sequence made of injectives ending at $M$:

$$M \longrightarrow I^1(M) \longrightarrow \cdots \longrightarrow I^n(M) \longrightarrow \Sigma M.$$

• Any admissible $n$-exact sequence

$$X^0 \xrightarrow{d^0} X^1 \xrightarrow{d^1} \dotsb \xrightarrow{d^n} X^{n+1}$$

induces a standard $(n+2)$-angle in the stable category

$$X^0 \xrightarrow{\overline{d^0}} \cdots \xrightarrow{\overline{d^n}} X^{n+1} \xrightarrow{\overline{d^{n+1}}} \Sigma X^0.$$

The axioms of an $(n+2)$-angulated category are satisfied because the construction "stabilizes" higher exactness via quotienting by projective-injective morphisms. This generalizes Happel's result for $n=1$ and triangulated categories.

The construction extends to stable categories of Frobenius $n$-exangulated categories (Liu et al., 2019), wherein $n$-exangulated structure is a further generalization (see below).

3. Relation to $n$-Exangulated Categories and Generalizations

$n$-exangulated categories, introduced by Herschend–Liu–Nakaoka, subsume both $n$-exact and $(n+2)$-angulated categories (Herschend et al., 2017). An $n$-exangulated category is an additive category $\mathcal{C}$ with a biadditive bifunctor $\mathbb{E}$ and a realization $s$, assigning to each extension class $\delta \in \mathbb{E}(C, A)$ a "distinguished" $(n+2)$-term complex (an $n$-exangle). $(n+2)$-angulated categories correspond precisely to those $n$-exangulated categories where $\mathbb{E}(C, A) = \mathcal{C}(C, \Sigma A)$ and $s$ is given by the assignment of $(n+2)$-angles (Herschend et al., 2017, He et al., 2020).

When working in the Frobenius $n$-exangulated context, the stable category is always $(n+2)$-angulated (Liu et al., 2019).

4. Axiomatic Streamlining and Morphism Axiom Redundancy

Within both $n$-angulated and $(n+2)$-angulated categories, it has been established that the morphism axiom ((N3): ability to extend a morphism between the bases of two angles to a morphism of angles) is implied by the other axioms, specifically the mapping cone (generalized octahedral) axiom and the existence (N1)(c) (Arentz-Hansen et al., 2016). This streamlines the construction and verification of $(n+2)$-angulated categories, as only existence, rotation, and higher octahedral conditions require direct checking.

5. Connections with $n$-Abelian, $n$-Exact, and Cluster Tilting Categories

Many naturally occurring $(n+2)$-angulated categories arise as stable categories of Frobenius $n$-exact categories, and $n$-exact categories themselves are generalized exact categories with "long" exact sequences of length $n+2$ (Jasso, 2014). Notably:

• $n$-cluster tilting subcategories of abelian or exact categories are always $n$-abelian or $n$-exact (Jasso, 2014).
• Passing to the stable category of a Frobenius $n$-exact (or $n$-exangulated) $n$-cluster tilting subcategory yields a canonical $(n+2)$-angulated category, placing higher Auslander–Reiten theory within this categorical setting (Jasso, 2014, Liu et al., 2019).

6. Applications and Examples

$(n+2)$-angulated categories have significant applications:

• Higher representation theory: $(n+2)$-angulated categories capture the homological structure of higher cluster categories, particularly those associated with $n$-representation finite and $n$-representation infinite algebras.
• Commutative ring theory and singularity theory: Categories of Cohen–Macaulay modules over Gorenstein isolated singularities admit Frobenius $n$-exact structures, whose stable categories become $(n+2)$-angulated (Jasso, 2014).
• Cluster tilting theory and mutation: $(n+2)$-angulated frameworks provide a natural context for mutation of tilting objects and the paper of maximal rigid subcategories.

7. Formulas and Structural Diagrammatics

Central formulas for $(n+2)$-angulated categories include:

• Admissible $n$-exact sequence in $\mathcal{M}$ (the “preimage”):

$$X^0 \xrightarrow{d^0} X^1 \xrightarrow{d^1} \cdots \xrightarrow{d^{n-1}} X^n \xrightarrow{d^n} X^{n+1}$$

• Induced standard $(n+2)$-angle in $\underline{\mathcal{M}}$:

$$X^0 \xrightarrow{\overline{d^0}} X^1 \xrightarrow{\overline{d^1}} \dotsb \xrightarrow{\overline{d^n}} X^{n+1} \xrightarrow{\overline{d^{n+1}}} \Sigma X^0$$

• Rotation (auto-equivalence):

$$X^1 \xrightarrow{\alpha^1} X^2 \to \cdots \to X^{n+1} \xrightarrow{\alpha^{n+1}} \Sigma X^0 \xrightarrow{(-1)^n \Sigma\alpha^0} \Sigma X^1$$

8. Theoretical Impact and Perspectives

The introduction of $(n+2)$-angulated categories, especially in connection with the stabilization of higher exact categories and the construction of higher cluster categories, has unified strands of higher homological algebra:

• It provides a categorical framework for phenomena not captured by triangulated or exact structures alone.
• New examples, such as the stable categories of Frobenius $n$-exangulated categories that are not $n$-exact, show the increased flexibility and reach of this structure (Liu et al., 2019).
• The simplified axiomatics (removal of the morphism axiom) enhances applications and categorical constructions (Arentz-Hansen et al., 2016).

Table: Core Axioms and Features of $(n+2)$-Angulated Categories

Axiom / Feature	Description (in terms of $(n+2)$-angles)	Reference
(N1) Existence/Closure	Trivial angles, closure under sums/summands, completions	(Jasso, 2014)
(N2) Rotation	Left-rotation invariance of $(n+2)$-angles	(Jasso, 2014)
(N3) Morphism	Extend morphism to angle morphism (redundant)	(Arentz-Hansen et al., 2016)
(N4) Mapping cone	Generalized octahedral compatibility	(Jasso, 2014)
Suspension	Auto-equivalence ($\Sigma$) extends angles cyclically	(Jasso, 2014)

The unification of higher homological algebra via $(n+2)$-angulated categories continues to drive research on categorical tilting, higher representation theory, and the structure of singularities. The flexibility of the definition and its realization as stable categories of Frobenius $n$-exact and $n$-exangulated categories illustrate its theoretical power and wide applicability.