Comonadic Approach to Pretorsion Theories

• The paper demonstrates how pseudo-comonads structure bihereditary pretorsion theories as coalgebras in semiexact categories.
• It leverages N-kernels and N-cokernels to guarantee short-exact sequences mediate torsion decompositions effectively.
• The approach extends to generalized pretorsion theories, where torsion and torsion-free components are defined by morphisms rather than objects.

The comonadic approach to pretorsion theories organizes and generalizes the study of torsion-like phenomena within semiexact categories—categories endowed with a distinguished closed ideal of morphisms and an abundance of kernels and cokernels relative to that ideal. By constructing pseudo-comonads in suitable 2-categories of semiexact categories, this framework captures bihereditary and arbitrary pretorsion theories as coalgebras (or pseudo-coalgebras), and extends naturally to a broader “generalized” notion where torsion and torsion-free components are morphisms, not just objects (Caviglia et al., 16 Jan 2026).

1. Semiexact Categories and Pretorsion Theories

A semiexact category $(C,\N)$ consists of a category $C$ equipped with a closed ideal $\N$ of morphisms—closed under composition and factorizations by isomorphisms—where every morphism admits an $\N$-kernel and $\N$-cokernel. For each $g:Y\to Z$, the universal map $\ker_\N(g)\xrightarrow{k}Y$ ensures that any $h:X\to Y$ with $g\circ h\in\N$ factors uniquely through $k$; dually for cokernels.

Pretorsion theories in this context are pairs $(\mathcal T,\mathcal F)$ of full replete subcategories such that their intersection $\mathcal Z := \mathcal T\cap\mathcal F$ mediates a torsion-like decomposition of objects via the ideal $\N$. Specifically, (T1) demands $\Mor_C(T,F)\subseteq\N$ for all $T\in\mathcal T, F\in\mathcal F$, and (T2) requires for each $X$ an $\N$-short-exact sequence $T^X\rightarrow X\rightarrow F^X$ with $T^X\in\mathcal T, F^X\in\mathcal F$, satisfying $\ell^X=\ker_\N(r^X)$ and $r^X=\coker_\N(\ell^X)$.

Hereditary pretorsion theories preserve kernels under $X\mapsto T^X$; cohereditary theories preserve cokernels under $X\mapsto F^X$; both together define bihereditary pretorsion theories.

2. Comonad Structure for Bihereditary Pretorsion Theories

The comonadic construction is performed within the 2-category $\ClIdl$ whose objects are semiexact categories $(C,\N)$, 1-cells are functors preserving $\N$-kernels and $\N$-cokernels, and 2-cells are natural transformations. The 2-functor $\Omega\colon\ClIdl\to\ClIdl$ acts as follows:

• On objects: $(C,\N)\mapsto(\Ses(C,\N),[\N])$, where $\Ses(C,\N)$ is the category of $\N$-short-exact sequences and $[\N]$ the ideal of null-isomorphism sequences.
• On 1-cells: $G:(C,\N)\to(D,\M)$ induces $\Omega(G)=G_*:\Ses(C)\to\Ses(D)$, acting componentwise on short-exact sequences.
• On 2-cells: Natural transformations $\alpha:G\Rightarrow H$ induce componentwise maps on $\Ses(C)$.

Counit $\varepsilon_C:\Ses(C)\to C$ selects the middle object of a short-exact sequence, and comultiplication $\delta_C:\Ses(C)\to\Ses(\Ses(C))$ utilizes the canonical pretorsion theory in $\Ses(C)$. The data $(\Omega, \delta, \varepsilon, \Xi)$, with an invertible modification $\Xi$, forms a pseudo-comonad on $\ClIdl$.

Normal pseudo-coalgebras for $\Omega$ correspond exactly to bihereditary pretorsion theories. The assignment $\lambda:(C,\N)\to\Ses(C)$ encodes for each $X\in C$ the canonical short-exact sequence representing the torsion and torsion-free decomposition, and the coalgebra axioms enforce the hereditary and cohereditary properties.

Main Theorem (Th. 3.8): The 2-category of normal pseudo-coalgebras for $(\Omega,\delta,\varepsilon,\Xi)$ is 2-isomorphic to the 2-category of bihereditary pretorsion theories.

3. Extension to General Pretorsion Theories and Generalized Concepts

To extend the framework from bihereditary to arbitrary pretorsion theories, one works with the 2-category $\ClIdl^{\rm ex}$ of semiexact categories and exact-sequence-preserving functors (not necessarily short exact). The pseudo-comonad $\Omega^{\mathrm{ex}}$ on $\ClIdl^{\mathrm{ex}}$ is defined so that $\Omega^{\mathrm{ex}}(G)$ for a 1-cell $G$ sends short exact sequences to chosen replacements of the corresponding exact sequence in the target category.

Normal pseudo-coalgebras for $\Omega^{\mathrm{ex}}$ select, for each $X$, an $\N$-short-exact sequence and encode corresponding factorization data for morphisms; however, only certain nullity properties are guaranteed—torsion and torsion-free are now characterized by their morphisms rather than the existence of specific classes of objects.

Genuine pretorsion theories are precisely those pseudo-coalgebras for which the functor $\lambda$ preserves the chosen short exact sequences for its own image; otherwise, one obtains a generalized concept of pretorsion theory where torsion decomposition does not necessarily arise from preselected classes in $C$.

Main Theorem (Th. 4.10): Every pretorsion theory determines a normal pseudo-coalgebra for $\Omega^{\mathrm{ex}}$; genuine pretorsion theories are identified by preservation properties under $\lambda$, while other pseudo-coalgebras capture generalized pretorsion theories.

4. Proof Strategies and Structural Insights

The construction leverages detailed calculations within the category of short exact sequences $\Ses(C)$. A morphism $(u,v,w):(X\to Y\to Z)\to(X'\to Y'\to Z')$ is the $[\N]$-kernel of another precisely when $u=\ker(u')$, $v=\ker(v')$ in $C$; similarly for cokernels.

Proposition 2.11 establishes the canonical bihereditary pretorsion theory on $\Ses(C)$: the torsion class $T$ consists of sequences $\bullet\iso\bullet\to\bullet$, and the torsion-free class $F$ of sequences $\bullet\to\bullet\iso\bullet$.

The counit $\varepsilon_C$ and the embedding $\delta_C$ show that $\Ses(C)$ is the cofree bihereditary pretorsion theory generated by $(C,\N)$.

Unwinding the pseudo-coalgebra axioms precisely retrieves the defining properties (T1), (T2), and the hereditary/cohereditary conditions for the corresponding pretorsion theory.

5. Illustrative Examples and Connections

Several examples clarify the scope and embedding of the comonadic framework:

• Rectangular torsion theories: Previous work by the authors [CJM-’25] analyzed “rectangular” torsion theories—where the canonical reflector is equivalent to a product $T\times F$—as pseudo-algebras for the “squaring” monad on pointed categories $\Cat_*$. This comonadic approach reinterprets these theories as pseudo-coalgebras for $\Omega$ on semiexact categories; all such rectangular theories are bihereditary.
• Finite preorders: For a finite totally ordered set $X = \{1 < 2 < \cdots < n\}$, assigning boundary points to opposite torsion classes and using the standard factoring-through-boundary condition yields a multitude of finite pretorsion theories, each bihereditary, and thus recognizable as coalgebras for $\Omega$.

6. Future Outlook and Generalized Pretorsion Theories

The $\Omega^{\mathrm{ex}}$ extension provides a systematic method for constructing all pretorsion theories and suggests a more expansive notion—where torsion and torsion-free constituents are morphisms and not necessarily objects. The framework anticipates further study of the adjoint string at each semiexact category: $\Ses(C) \leftrightarrow C \leftrightarrow \bullet = \bullet \to \bullet \leftrightarrow \bullet \to \bullet = \bullet$ as well as analysis of the monadic analogue for the rectangular case. This suggests new avenues for both classification and application of torsion phenomena beyond the traditional construal, including categorical structures with adjunctions and generalized decompositions (Caviglia et al., 16 Jan 2026).