Bihereditary Pretorsion Theories

• Bihereditary pretorsion theories are a generalization of classical torsion theories, defined by a pair of subcategories with stability under subobjects and quotients.
• They employ universal constructions, closure operators, and factorization systems to ensure functorial preservation in semiexact and coherent categories.
• The 2-categorical, comonadic characterization provides a unified framework for analyzing torsion phenomena and advancing non-classical homological algebra.

A bihereditary pretorsion theory generalizes classical torsion theories by relaxing pointedness and working relative to a chosen class of “trivial” objects or morphisms. In this context, biheredity refers to a refined stability property: the torsion class is stable under subobjects and the torsion-free class under quotients, both understood in a relative, often categorical, sense. The subject connects categorical algebra, factorization systems, closure operators, and comonadic homological algebra, and has received substantial recent theoretical development, including explicit 2-dimensional comonadic characterizations and universal construction techniques (Caviglia et al., 16 Jan 2026, Mantovani et al., 8 Oct 2025, Borceux et al., 2022).

1. Foundations: Semiexact and Coherent Categories

Bihereditary pretorsion theories fundamentally rely on semiexact or prenormal categorical frameworks. A semiexact category $(\C,\N)$ consists of a category $\C$ and a closed ideal $\N$ of “null morphisms”, which is stable under composition as well as (relative) kernels and cokernels: every morphism has an $\N$-kernel and $\N$-cokernel, and any kernel or cokernel of a null morphism is an isomorphism. Objects are null if their identity lies in $\N$ (Caviglia et al., 16 Jan 2026).

In the context of coherent categories, there are well-structured lattices of subobjects with distributivity and good pullback properties. Distinguished subobjects—those satisfying certain axioms (including closure under unions and pullbacks)—play a central role in many universal constructions (Borceux et al., 2022).

In more general prenormal categories (as in Mantovani–Messora), triviality is specified by a class $Z$ of full, replete subcategories. Relative notions, such as $Z$-kernels and $Z$-cokernels, substitute for their classical counterparts (Mantovani et al., 8 Oct 2025).

2. Pretorsion Theories: Definition and Structure

Let $(\C,\N)$ be semiexact. A pretorsion theory is a pair of full, replete subcategories $(\T, \F)$ with intersection $\Z = \T\cap\F$, such that:

• $(T1)$ All morphisms from $\T$ to $\F$ are null, i.e., $\Hom_\C(T,F)\subseteq \N_\Z$.
• $(T2)$ Every object $C$ fits into a short $\N_\Z$-exact sequence $T_C \to C \to F_C$, with $T_C\in\T$, $F_C\in\F$, where the maps are (relative) kernel and cokernel (Caviglia et al., 16 Jan 2026, Borceux et al., 2022).

In prenormal categories, these data appear with respect to a class $Z$ of trivial objects: a $Z$-exact sequence is defined in terms of $Z$-kernels and $Z$-cokernels, and the category admits a “pretorsion decomposition” for each object (Mantovani et al., 8 Oct 2025).

3. Hereditary, Cohereditary, and Bihereditary Structure

A pretorsion theory $(\T,\F)$ is:

• Hereditary if the coreflector $t\colon \C\to\T$ preserves (relative) kernels (or, in the relative sense, $T$ is closed under subobjects defined by $Z$-normal monos).
• Cohereditary if the reflector $f\colon \C\to\F$ preserves (relative) cokernels ($F$ is closed under quotients defined by $Z$-normal epis).
• Bihereditary if both conditions hold (Caviglia et al., 16 Jan 2026, Mantovani et al., 8 Oct 2025, Borceux et al., 2022).

Several characterizations are available:

• In terms of closure operators: The closure operators induced by the reflector/coreflector on the classes of normal monos/epis are both hereditary if and only if the pretorsion theory is bihereditary (Mantovani et al., 8 Oct 2025).
• Functorial preservation: For $(\T,\F)$ in a prenormal category, biheredity is equivalent to the coreflector preserving $Z$-normal monos and the reflector preserving $Z$-normal epis.
• Factorization systems: A characteristic pretorsion theory induces an orthogonal factorization system; biheredity corresponds to pullback/pushout stability of the classes (Mantovani et al., 8 Oct 2025).

4. Comonadicity and 2-Categorical Perspective

A pivotal development is the 2-categorical, comonadic characterization of bihereditary pretorsion theories. In the 2-category $\mathbf{SExact}$ (objects: semiexact categories; 1-cells: functors preserving $\N$-kernels/cokernels; 2-cells: natural transformations), the endo-2-functor

$$\mathbf{Ses}\colon \mathbf{SExact} \to \mathbf{SExact}$$

assigns to each $(\C,\N)$ the category of short exact sequences $\Ses(\C)$, with an induced ideal of null morphisms. This functor underlies a normal pseudo-comonad $(\mathbf{Ses},\epsilon,\delta,\Xi)$ with explicitly described counit and comultiplication, encapsulating the "universal short exact sequence" construction (Caviglia et al., 16 Jan 2026).

A normal pseudo-coalgebra for $\mathbf{Ses}$ consists of a functorial choice of short exact sequence for each object, compatible with coherence isomorphisms. The fundamental theorem establishes:

• The 2-category of normal pseudo-coalgebras for $\mathbf{Ses}$ is 2-equivalent, over $\mathbf{SExact}$, to the 2-category of bihereditary pretorsion theories and kernel-cokernel-preserving functors.

Thus, bihereditary pretorsion theories correspond precisely to those semiexact categories $\C$ equipped with a splitting $\lambda\colon \C\to\Ses(\C)$ of the universal short-exact-sequence comonad.

5. Universal Constructions and Bihereditary Hulls

In coherent categories, universal constructions yield bihereditary closures of arbitrary pretorsion theories. The stable torsion theory $\Sigma(\C)$, constructed using partial maps and congruences, is universal among torsion functors respecting distinguished unions. Its dual, the co-stable theory $\Gamma(\C)$, is universal among functors preserving distinguished intersections and torsion-free structure (Borceux et al., 2022). Both produce bihereditary (hereditary and cohereditary) torsion theories.

These hulls are related by adjoint-reflective/coreflective inclusions: $\Sigma(\C) \xhookrightarrow{} \C \xhookleftarrow{} \Gamma(\C)$ demonstrating how any pretorsion theory naturally interpolates between its hereditary and cohereditary “envelopes”.

6. Examples and Applications

Instances of bihereditary pretorsion theories span diverse algebraic and categorical settings:

• Abelian and semi-abelian categories: Classical (pointed) hereditary torsion theories and Serre subcategories are encompassed as special cases.
• Commutative monoids: The pair (torsion monoids, torsion-free monoids) forms a bihereditary pretorsion theory (Mantovani et al., 8 Oct 2025).
• Inverse commutative monoids: With idempotent monoids as trivial, the torsion-free subcategory yields another bihereditary example.
• Topoi: With appropriate distinguished subobjects (e.g., clopen subobjects), one recovers stable categories corresponding to preorders, again yielding bihereditarity (Borceux et al., 2022).
• Groupoids: The pair (connected groupoids, totally disconnected groupoids) in the category of groupoids, with discrete groupoids as trivial, is bihereditary.

Universal constructions in these contexts provide “bihereditary hulls” of general pretorsion theories, ensuring closure under subobjects and quotients in the appropriate relative sense.

7. Structural Significance and Research Directions

Bihereditary pretorsion theories unify perspectives from torsion theory, closure operators, and factorization systems. The comonadic approach consolidates their structure in the setting of 2-dimensional homological algebra, delivering canonical “cofree” extensions and enabling uniform treatment of derived and short-exact-sequence categories (Caviglia et al., 16 Jan 2026).

This framework illuminates the interconnections between different types of torsion phenomena, their stability properties, and categorical approaches to exactness. The explicit equivalence between bihereditarity and comonadicity not only characterizes these theories but also suggests new tools for investigating change-of-torsion functors, categorical derived functors, and the global landscape of structural closure in algebra, topos theory, and beyond.

Further development includes the refinement of factorization systems, the study of pretorsion in non-pointed or relative settings, and applications to the structure of stable and co-stable categories (Mantovani et al., 8 Oct 2025, Borceux et al., 2022). The categorical methods outlined establish a foundation for ongoing research in non-classical homological algebra and the theory of derived categorical structures.