d-Abelian Categories

• d-Abelian categories are additive, idempotent complete structures where every morphism admits a d-kernel and a d-cokernel, forming d-exact sequences.
• They generalize classical abelian categories (the d=1 case) and unify module theory, categorical, and model-theoretic perspectives via precise axioms.
• Equivalences with d-cluster tilting subcategories and links to (d+2)-angulated categories underpin applications in higher Auslander–Reiten theory and homological decomposition.

A $d$-abelian category is a higher-homological analogue of an abelian category, defined via axioms that generalize the role of kernels, cokernels, and exact sequences to “$d$-kernels”, “$d$-cokernels”, and “$d$-exact sequences” of length $d+2$. This notion formalizes structures observed in $d$-cluster tilting subcategories—critical in modern higher representation theory. The $d$-abelian framework not only encompasses abelian categories (the case $d=1$), but also brings together module-theoretic, categorical, and model-theoretic perspectives and enables explicit constructions and equivalences central to higher Auslander–Reiten theory and related areas.

1. Core Definitions and Structural Axioms

A $d$-abelian category is an additive, idempotent complete category $\mathcal{M}$ in which every morphism has both a $d$-kernel and a $d$-cokernel, and whose monomorphisms and epimorphisms extend to $d$-exact sequences. Specifically, for fixed $d \geq 1$, the following hold:

• $d$-kernel of $f^{d}: X^{d} \to X^{d+1}$: A sequence

$X^{0} \xrightarrow{f^0} X^{1} \to \cdots \to X^{d}$

such that, for every $B \in \mathcal{M}$, the induced sequence

$$0 \to \mathrm{Hom}(B,X^{0}) \to \cdots \to \mathrm{Hom}(B,X^{d+1})$$

is exact.

• $d$-cokernel of $f^0: X^0 \to X^1$: A sequence

$X^1 \xrightarrow{f^1} X^2 \to \cdots \to X^{d+1}$

such that, for every $B \in \mathcal{M}$, the induced sequence

$$\mathrm{Hom}(X^{d+1},B) \to \cdots \to \mathrm{Hom}(X^{0},B) \to 0$$

is exact.

• A morphism $f^0$ (resp. $f^{d}$) is monic (resp. epic) if the above induced maps are injective (resp. surjective) in the extremal degrees.
• $d$-exact sequence: A complex

$0 \to A^{0} \to A^{1} \to \cdots \to A^{d+1} \to 0$

whose initial $d$ morphisms form a $d$-kernel of $A^{d} \to A^{d+1}$ and last $d$ morphisms form a $d$-cokernel of $A^{0} \to A^{1}$.

A $d$-abelian category satisfies the following axioms (as in (Jasso, 2014, Fedele, 2018)):

1. (A0) Idempotent completeness.
2. (A1) Existence of $d$-kernels and $d$-cokernels for every morphism.
3. (A2) Any monomorphism $f^0$ and any choice of $d$-cokernel of $f^0$ yield a $d$-exact sequence; dually for epimorphisms.

These axioms generalize the categorical structure of abelian categories: when $d=1$, the notions collapse to the familiar kernel-cokernel-exact sequence framework.

2. Equivalence with $d$-Cluster Tilting Subcategories

$D$-abelian categories were axiomatized to capture the essential properties of $d$-cluster tilting subcategories in abelian categories. For a full subcategory $X$ of an abelian category $\mathcal{A}$ to be $d$-cluster tilting, it must be functorially finite, generating and cogenerating, and satisfy: $$X = \{A \in \mathcal{A} \mid \forall 1 \leq i \leq d-1,\; \mathrm{Ext}_{\mathcal{A}}^{i}(X, A) = 0\} = \{A \in \mathcal{A} \mid \forall 1 \leq i \leq d-1,\; \mathrm{Ext}_{\mathcal{A}}^{i}(A, X) = 0\}$$

(Jasso, 2014, Kvamme, 2020, Kvamme, 2016). Such subcategories are always $d$-abelian (Jasso, 2014), and under broad circumstances—particularly when projectively generated—every $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category (Kvamme, 2016, Kvamme, 2020). Moreover, recent advances remove projective generation restrictions: any $d$-abelian category arises as a $d$-cluster tilting subcategory up to equivalence (Kvamme, 2020). Thus $d$-abelian categories provide a precise axiomatization for $d$-cluster tilting theory.

3. Examples and Explicit Constructions

A canonical source of $d$-abelian categories is as follows. Let $\Lambda$ be an $n$-representation-finite algebra (e.g., a suitable quotient of a path algebra), with a unique $d$-cluster tilting $\Lambda$-module $M$. Then $\mathrm{add}\, M$ is a $d$-abelian category. For $d=2$, one may take Cohen–Macaulay modules over an isolated singularity (e.g., invariant subrings of polynomial rings), where $\mathrm{add}\, S$ is $2$-abelian if $S$ is the regular ring (Jasso, 2014).

Construction methods also generalize to derived and angulated categories. For instance, for a hereditary abelian category $\mathcal{C}$, the subcategory

$$\mathcal{C}[0,m] = \{ X \in D^{b}(\mathcal{C}) \;|\; H_{*}(X) \text{ concentrated in } 0 \leq j \leq m \}$$

is $d$-abelian for $d=3m+1$ (Jorgensen et al., 22 Sep 2025). More generally, for an $(n+2)$-angulated category $\mathcal{T}$ built from an $n$-cluster tilting object $M$, one obtains a $d$-abelian subcategory

$$\mathcal{T}[0,m] = \operatorname{add} \bigoplus_{j=0}^m \Sigma_n^j M$$

with $d = (n+2)(m+1) - 2$ (Jorgensen et al., 22 Sep 2025). These constructions yield $d$-abelian categories with additional features determined by the ambient category; e.g., the existence of enough injectives or products depends on the hereditary or linearity properties of $\mathcal{C}$ (Jorgensen et al., 22 Sep 2025).

4. Relationship to $(d+2)$-Angulated Categories and Higher Auslander–Reiten Theory

$D$-abelian categories and $(d+2)$-angulated categories are deeply connected. The stable category of a Frobenius $d$-exact category acquires a canonical structure of a $(d+2)$-angulated category, whose distinguished $(d+2)$-angles generalize distinguished triangles (Jasso, 2014): $X^0 \to X^1 \to \cdots \to X^{d+1} \to \Sigma X^0$

A $d$-abelian category gives rise to a $d$-exact category whose stable quotient (modulo projective-injective objects) acquires a natural higher angulated structure. These connections generalize the classical passage from abelian categories and their derived (triangulated) categories to the $d$-abelian context (Jacobsen et al., 2017, Fedele, 2018). Explicitly, for a cluster tilting object $T$ in an appropriate $(d+2)$-angulated category $\mathcal{T}$, the quotient $\mathcal{T}/I$ (with $I$ the ideal of morphisms factoring through $\operatorname{add} T$) is $d$-abelian and equivalent to a $d$-cluster tilting subcategory in module categories over $\mathrm{End}_{\mathcal{T}}(T)$ (Jacobsen et al., 2017).

There are deep links to higher Auslander–Reiten theory, notably:

• Generalizations of Auslander–Reiten sequences to $(d+2)$-angles (Auslander–Reiten $(d+2)$-angles) in $(d+2)$-angulated categories, with strong correspondences and covering properties (Fedele, 2018).
• Bijections between wide subcategories of $d$-abelian and their associated angulated categories, supporting classification results and categorical mutations.

5. Model Theory, Definable Categories, and Functorial Aspects

$D$-abelian categories—and more generally, definable additive categories—arise as “categories of models” of coherent additive theories expressed via pp-formulas (Prest, 2012). There exists a close anti-equivalence between the 2-category of small abelian categories, the 2-category of definable additive categories with interpretation functors (functors commuting with products and direct limits), and locally coherent Grothendieck categories with coherent morphisms (Prest, 2012). A definable category is a full subcategory of a module category closed under products, direct limits, and pure subobjects. These categories capture the behavior of additive theories in both algebraic and model-theoretic settings, and the functor category $\mathrm{fun}(D)$—the category of exact functors into $\mathbf{Ab}$—recovers the small abelian category as the language of pp-imaginaries. The “free abelian category” construction provides a canonical bridge: for a skeletally small preadditive category $R$, the abelian envelope $\mathrm{Ab}(R)$ admits a universal embedding and factorization property.

Key applications include Morita invariance in module and model theory, geometric analogues (“structure sheaves” of definable additive categories), as well as dualities between module categories and functor categories (Prest, 2012).

6. Completion, Universality, and Categorical Decomposition

The Ind-completion of a small $d$-abelian category $\mathcal{M}$, $\mathrm{Ind}(\mathcal{M})$, plays a prominent role in passing from “small” to “big” $d$-abelian categories and exploring functor categories (Ebrahimi et al., 2022): $$\mathrm{Ind}(\mathcal{M}) \cong \mathcal{L}_d(\mathcal{M})$$ where $\mathcal{L}_d(\mathcal{M})$ denotes the subcategory of left $d$-exact functors from $\mathcal{M}^{op}$ into $\mathbf{Ab}$. The completion $\mathrm{Ind}(\mathcal{M})$ is a generating-cogenerating functorially finite subcategory of $\mathrm{Mod}\, \mathcal{M}$ and satisfies all $d$-cluster tilting properties except, in general, $d$-rigidity. The $d$-rigidity of $\mathrm{Ind}(\mathcal{M})$ is decisive for it to be $d$-cluster tilting (affirmed for $d=2$ under finiteness constraints via cotorsion theory). The existence and uniqueness of such completions are central to understanding extension of $d$-abelian categories to environments with products, coproducts, and filtered colimits.

7. Open Problems and Directions

Despite strong structural results, several challenges remain:

• Intrinsic characterization: A direct, easily checkable list of intrinsic, purely categorical conditions for definable (and thus $d$-abelian) categories, not relying on module-theoretic embeddings, is still lacking (Prest, 2012).
• Rigidity of completions: The $d$-rigidity of $\mathrm{Ind}(\mathcal{M})$ for general $\mathcal{M}$ is an open question, with implications for when completions are genuinely $d$-cluster tilting (Ebrahimi et al., 2022).
• Extension properties: Questions persist on detecting properties of a definable category from its pure-injective objects or via restriction of functorial operations (Prest, 2012).
• Geometric and derived analogies: The full extent to which the geometric “structure sheaf” and quotient categories in the abelian setting analogize scheme theory, and subtleties regarding closed subschemes, require further investigation.
• Decomposition and recollement in higher settings: The blueprint provided by orthogonal and derived decompositions in abelian categories invites the search for $d$-exact analogues, with expectations for higher stratifications and recollements (Chen et al., 2018).

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In summary, $d$-abelian categories encapsulate higher analogues of abelian categorical structure, tightly interlinking cluster tilting, module theory, higher angulated categorical structures, and model theory, and display a wealth of internal and external equivalences. While the categorical and homological framework is highly developed, ongoing challenges focus on intrinsic characterizations, extension properties of completions, and the link to generalized algebraic geometry and decomposition theory.