Cofinally Closed Monobricks in Abelian Categories

• Cofinally closed monobricks are maximal sets of bricks that intrinsically classify torsion-free classes in length abelian categories.
• They establish a bijective correspondence between monobricks and left Schur subcategories, unifying torsion theory and the study of wide subcategories.
• Their construction via recollements and combinatorial enumeration offers a robust framework for analyzing module filtrations and torsion lattices.

A cofinally closed monobrick is a maximal set of bricks in a length abelian category with respect to the injection relation, providing an intrinsic combinatorial classification of torsion-free classes and a unified approach to torsion theory and wide subcategories. The theory of cofinally closed monobricks has been developed through several foundational works, notably by Enomoto and collaborators, and extends to contexts involving recollements and module categories of finite-dimensional algebras (Enomoto, 2020, Ringel, 27 Nov 2024, Yang et al., 6 Feb 2025, Zhang et al., 18 Oct 2025).

1. Definitions: Bricks, Monobricks, and Cofinal Closure

Let $\mathcal{A}$ be a length abelian category. An object $X \in \mathcal{A}$ is a brick if its endomorphism ring $\operatorname{End}_{\mathcal{A}}(X)$ is a division ring. Bricks generalize simple objects and may include more complex indecomposables in nonlocal settings.

A subset $$\mathcal{M} \subset \operatorname{Brick}(\mathcal{A})$$ is a monobrick if for any $X, Y \in \mathcal{M}$, every nonzero morphism $f: X \to Y$ is a monomorphism. Unlike semibricks (where every nonzero morphism is an isomorphism), monobricks allow a richer class of morphism structures, provided nonzero maps remain injective.

Given the partial order $X \leq Y$ iff there exists a monomorphism $X \to Y$, one defines a cofinal extension: a monobrick $\mathcal{N}$ is a cofinal extension of $\mathcal{M}$ if $\mathcal{M} \subseteq \mathcal{N}$ and for every $N \in \mathcal{N}$, there exists $M \in \mathcal{M}$ with an injection $N \hookrightarrow M$.

The cofinal closure $\overline{\mathcal{M}}$ of $\mathcal{M}$ is the union of all its cofinal extensions. The monobrick $\mathcal{M}$ is cofinally closed if $\overline{\mathcal{M}} = \mathcal{M}$. Equivalently, the condition (CC) holds:

For any $N \in \operatorname{Brick}(\mathcal{A})$ with an injection $N \to M$ for some $M \in \mathcal{M}$, if $N \notin \mathcal{M}$, then there exists a nonzero non-injective map $N \to M'$, $M' \in \mathcal{M}$.

This condition ensures that $\mathcal{M}$ is maximal with respect to the injection structure and cannot be properly enlarged without violating the monobrick property (Enomoto, 2020).

2. Structural Role: Bijections and Classification

A critical result is the bijective correspondence between monobricks (resp. cofinally closed monobricks, semibricks) and left Schur subcategories (resp. torsion-free classes, wide subcategories) in $\mathcal{A}$ (Enomoto, 2020, Zhang et al., 18 Oct 2025).

Left Schur subcategories $\mathcal{E} \subseteq \mathcal{A}$ are extension-closed subcategories where every simple object in $\mathcal{E}$ has the property that every map from it to any object in $\mathcal{E}$ is either zero or an injection.

The correspondence is provided by the mutually inverse maps:

• Given a left Schur subcategory $\mathcal{E}$: $\operatorname{sim}(\mathcal{E})$ is the set of simple objects (which forms a monobrick).
• Given a monobrick $\mathcal{M}$: $\mathrm{Filt}(\mathcal{M})$ is the full subcategory of objects with filtrations whose factors lie in $\mathcal{M}$.

Symbolically, for subcategory classes: $$\begin{aligned} &\{\text{Schur}_L\ \mathcal{A}\} \;\xleftrightarrow{\sim}\; \{\operatorname{mbrick}\,\mathcal{A}\} \ &\{\text{torf}\,\mathcal{A}\} \;\xleftrightarrow{\sim}\; \{\operatorname{mbrick}_{\mathrm{c.c.}}\,\mathcal{A}\} \ &\{\text{wide}\,\mathcal{A}\} \;\xleftrightarrow{\sim}\; \{\operatorname{sbrick}\,\mathcal{A}\} \end{aligned}$$ where $\operatorname{mbrick}_{\mathrm{c.c.}}\,\mathcal{A}$ are cofinally closed monobricks (Zhang et al., 18 Oct 2025).

The significance is that torsion-free classes in $\mathcal{A}$ are classified by cofinally closed monobricks, providing a direct combinatorial invariant for torsion theory in length abelian categories (Enomoto, 2020).

3. Filtrations and Directedness: Connection to Module Structure

Bricks are the atomic modules in the representation theory of artin algebras (Ringel, 27 Nov 2024). Filtrations by bricks (i.e., chains $0 = M_0 \subset M_1 \subset \cdots \subset M_m = M$ with $M_i/M_{i-1}$ bricks) encode the decomposition and structure of modules. In the context of brick chains (ordered sequences $(B_1, \dots, B_m)$ with $\operatorname{Hom}(B_i, B_j) = 0$ for $i < j$), module filtrations are controlled in a “directed” fashion.

The set of “top bricks” in the filtration—uniquely associated to a module via an iterated endotop construction—encapsulates the position of the module within the lattice of torsion classes. This ordered set often manifests as a (cofinally closed) monobrick, which acts as a minimal collection of bricks characterizing the module’s position in the torsion-theoretic structure (Ringel, 27 Nov 2024).

This framework provides an alternative to $\tau$-tilting theory for understanding the composition of module categories, emphasizing the brickwise, directed behavior underpinning torsion-theoretic lattices.

4. Gluing via Recollements: Construction and Stability

Given a recollement of length abelian categories: $$\mathcal{Y} \xrightarrow{i_*} \mathcal{X} \xrightarrow{j^*} \mathcal{Z},$$ the construction of cofinally closed monobricks in $\mathcal{X}$ from those in $\mathcal{Y}$ and $\mathcal{Z}$ is explicit (Zhang et al., 18 Oct 2025): $$\mathcal{M}_{\mathcal{X}} := i_*(\mathcal{M}_{\mathcal{Y}}) \sqcup j_*(\mathcal{M}_{\mathcal{Z}})$$ where $$\mathcal{M}_{\mathcal{Y}} \in \operatorname{mbrick}_{\mathrm{c.c.}}(\mathcal{Y})$$ and $$\mathcal{M}_{\mathcal{Z}} \in \operatorname{mbrick}_{\mathrm{c.c.}}(\mathcal{Z})$$. The resulting set is a cofinally closed monobrick in $\mathcal{X}$ under assumptions such as the exactness of $i^!$.

In categorical terms, the glued left Schur subcategory is: $$\mathcal{E}_{\mathcal{X}} := \{ X \in \mathcal{X} \mid j^*X \in \mathcal{E}_{\mathcal{Z}},\ i^! X \in \mathcal{E}_\mathcal{Y} \},$$ and the monobrick of simples is as above. This construction preserves the relevant properties (extension-closure, cofinal closure) required for the correspondence with torsion-free classes and ensures compatibility with the underlying recollement structure (Yang et al., 6 Feb 2025, Zhang et al., 18 Oct 2025).

This methodology is applicable in module categories over triangular matrix algebras and other settings, enabling the explicit synthesis of torsion-free classes (and hence cofinally closed monobricks) from simpler categories.

5. Finiteness Criteria and Combinatorial Enumeration

The number and structure of cofinally closed monobricks—and hence torsion-free classes—are closely dictated by the brick structure of the category. A finiteness theorem states that there exist only finitely many torsion-free classes (and wide subcategories) if and only if $\mathcal{A}$ has finitely many bricks (Enomoto, 2020).

For Nakayama algebras, enumeration becomes highly combinatorial: bricks correspond bijectively to certain arc diagrams, and the count of monobricks is given explicitly by the large Schröder numbers: $$\#\,\operatorname{mbrick}\,A_n = \sum_{i=0}^{n-1} \binom{n}{i} \binom{n}{i+1}$$ establishing a powerful bridge between representation theory and enumerative combinatorics (Enomoto, 2020).

6. Implications and Unification of Subcategory Theory

The notion of cofinally closed monobricks unifies the following:

• Torsion-free classes $\longleftrightarrow$ cofinally closed monobricks.
• Wide subcategories $\longleftrightarrow$ semibricks.
• Left Schur subcategories $\longleftrightarrow$ monobricks in general.

This provides a single combinatorial paradigm for understanding the major torsion-theoretic and wide subcategories arising in length abelian categories and their module-theoretic counterparts (Enomoto, 2020, Zhang et al., 18 Oct 2025). The gluing and extension results in recollements further cement the robustness and transferability of this framework across complex categorical landscapes (Yang et al., 6 Feb 2025, Zhang et al., 18 Oct 2025).

7. Further Directions

The paper of cofinally closed monobricks informs ongoing research in:

• Precise behavior of torsion(-free) classes and lattices under recollement and derived functors.
• Extension to broader classes of abelian and triangulated categories.
• Interaction with ICE-closed subcategories, epibricks, and the applications of closure operations beyond classical settings (Yang et al., 6 Feb 2025).
• Enhanced combinatorial and enumerative approaches, especially in connection with $\tau$-tilting theory and other invariants.
• Explicit construction and classification in important families of algebras (e.g., triangular matrix algebras, Nakayama algebras).

This theory offers both structural unification and practical mechanisms for constructing and classifying abelian subcategories in algebraic contexts.