Extriangulated Factorization Systems

• Extriangulated factorization systems are frameworks defined on extriangulated categories that generalize exact and triangulated settings through cotorsion and s-torsion pair correspondences.
• They employ admissible weak factorization systems and orthogonality of cones to relate inflations, deflations, and morphism factorizations, ensuring cancellation and compatibility.
• This approach unifies model structures across exact, abelian, and triangulated contexts, opening avenues for constructing Hovey triples and exploring n-exangulated analogues.

Extriangulated factorization systems are factorization-theoretic structures defined on extriangulated categories, a framework introduced by Nakaoka–Palu that simultaneously generalizes exact and triangulated categories. In recent work, the expression is used in two closely related senses. One sense concerns admissible weak factorization systems, where the left and right classes are tied to inflations and deflations and are in bijection with cotorsion pairs; the other concerns inflation/deflation factorization systems defined by orthogonality of cones using both $\mathcal{C}$ and $\mathbb{E}^{-1}$, and these are in bijection with $s$-torsion pairs. Together, these developments organize cotorsion-theoretic and torsion-theoretic data into morphism factorizations intrinsic to the extriangulated setting (Ma et al., 2024, Xu et al., 6 Jul 2025).

1. Extriangulated framework

An extriangulated category is a triple $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$ in which $\mathcal{C}$ is an additive category, $$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$ is an additive bifunctor, and $\mathfrak{s}$ assigns to each $\mathbb{E}$-extension $\delta \in \mathbb{E}(C,A)$ an equivalence class

$$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$

called a conflation or extriangle. The morphism $\mathbb{E}^{-1}$0 is an inflation, $\mathbb{E}^{-1}$1 is a deflation, $\mathbb{E}^{-1}$2 is the cone of $\mathbb{E}^{-1}$3, and $\mathbb{E}^{-1}$4 is the cocone of $\mathbb{E}^{-1}$5 (Ma et al., 2024).

The ambient formalism supplies the exactness and gluing properties needed for factorization theory. The long exact sequences associated to extriangles control $\mathbb{E}^{-1}$6-morphisms and $\mathbb{E}^{-1}$7-extensions, while axiom (ET4) and its dual provide pushout, pullback, and composition behavior for extriangles. In the admissible weak factorization system setting, weak idempotent completeness, expressed as condition (WIC), is used to cancel inflations and deflations: if $\mathbb{E}^{-1}$8 is an inflation then $\mathbb{E}^{-1}$9 is an inflation, and if $s$0 is a deflation then $s$1 is a deflation (Ma et al., 2024).

A second layer of structure appears in extriangulated categories with negative first extensions. Such a category is equipped with an additive bifunctor

$s$2

and natural transformations compatible with every $s$3-triangle in such a way that exact sequences extend one step to the left. Exact categories and triangulated categories are examples. This additional datum is essential for the orthogonality notion used in the 2025 theory of inflation factorization systems (Xu et al., 6 Jul 2025).

A common source of confusion is that the two theories do not impose the same axioms. The admissible weak factorization system approach studies pairs of classes of all morphisms satisfying lifting and factorization, then restricts them by admissibility to inflations and deflations. The $s$4-torsion-pair approach studies pairs of classes of inflations or deflations directly, with orthogonality defined by vanishing between cones or cocones (Ma et al., 2024, Xu et al., 6 Jul 2025).

2. Admissible weak factorization systems and cotorsion pairs

In a category $s$5, a weak factorization system (WFS) is a pair $s$6 of classes of morphisms such that

$s$7

and every morphism factors as $s$8 with $s$9 and $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$0. The classes are closed under composition, retracts, and contain all isomorphisms. In the extriangulated context, an admissible weak factorization system (AWFS) is a WFS for which a morphism $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$1 lies in $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$2 if and only if $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$3 is an inflation and $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$4 lies in $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$5, and $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$6 lies in $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$7 if and only if $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$8 is a deflation and $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$9 lies in $\mathcal{C}$0. In particular, $\mathcal{C}$1 consists of inflations and $\mathcal{C}$2 consists of deflations (Ma et al., 2024).

For a subcategory $\mathcal{C}$3, the paper defines

$\mathcal{C}$4

and

$\mathcal{C}$5

These are the morphism classes associated to object classes in the main correspondence.

A cotorsion pair is a pair $\mathcal{C}$6 of full additive subcategories, closed under isomorphisms and direct summands, such that

$\mathcal{C}$7

and every object $\mathcal{C}$8 admits approximation extriangles

$\mathcal{C}$9

with $$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$0 and $$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$1. The orthogonality notation is

$$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$2

The central theorem establishes a bijection

$$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$3

between cotorsion pairs and admissible weak factorization systems. Explicitly,

$$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$4

and conversely

$$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$5

The theorem shows that $$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$6 and that the approximation extriangles are recovered from the factorizations supplied by the AWFS (Ma et al., 2024).

The proof uses three ingredients. First, factorization is obtained by combining approximation extriangles for cones or cocones with the octahedral-type behavior encoded in (ET4) and its dual. Second, lifting is detected by a lemma asserting that if every inflation with cone in $$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$7 lifts against a deflation $$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$8, then $$\mathbb{E}: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{Ab}$$9; dually, lifting against deflations with cocone in $\mathfrak{s}$0 forces cones in $\mathfrak{s}$1. Third, WIC is used to ensure that the pieces of the factorization remain inflations and deflations of the expected type.

3. Heredity, cancellation, and compatibility

A cotorsion pair $\mathfrak{s}$2 is hereditary if $\mathfrak{s}$3 is closed under cocones of deflations and $\mathfrak{s}$4 is closed under cones of inflations. The paper adopts this closure formulation and notes that the two conditions are equivalent. These closure properties translate into cancellation properties for the corresponding admissible weak factorization system (Ma et al., 2024).

For morphism classes $\mathfrak{s}$5 and $\mathfrak{s}$6, the relevant cancellation conditions are:

• Left cancellation: if $\mathfrak{s}$7 and $\mathfrak{s}$8, then $\mathfrak{s}$9.
• Right cancellation: if $\mathbb{E}$0 and $\mathbb{E}$1, then $\mathbb{E}$2.

The paper proves that a class $\mathbb{E}$3 is closed under cocones of deflations if and only if $\mathbb{E}$4 has left cancellation, and dually a class $\mathbb{E}$5 is closed under cones of inflations if and only if $\mathbb{E}$6 has right cancellation. Moreover, for the AWFS arising from a cotorsion pair, left cancellation for $\mathbb{E}$7 is equivalent to right cancellation for $\mathbb{E}$8.

This yields the equivalence

$\mathbb{E}$9

The theorem generalizes a result of Di–Li–Liang from abelian categories to extriangulated categories. In the abelian case, it recovers the criterion that a pair $\delta \in \mathbb{E}(C,A)$0 is complete and hereditary if and only if

$\delta \in \mathbb{E}(C,A)$1

is a WFS with the corresponding cancellation properties.

The same paper also studies compatibility. Two WFS $\delta \in \mathbb{E}(C,A)$2 and $\delta \in \mathbb{E}(C,A)$3 are compatible if three conditions hold: $\delta \in \mathbb{E}(C,A)$4, equivalently $\delta \in \mathbb{E}(C,A)$5; the class $\delta \in \mathbb{E}(C,A)$6 satisfies a 2-out-of-3 condition for composition; and if $\delta \in \mathbb{E}(C,A)$7, $\delta \in \mathbb{E}(C,A)$8, and $\delta \in \mathbb{E}(C,A)$9, then $$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$0. Two cotorsion pairs are compatible if

$$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$1

equivalently via a thick subcategory $$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$2 with

$$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$3

Theorem 3.22 identifies compatibility of cotorsion pairs with compatibility of the corresponding admissible weak factorization systems, again extending an abelian theorem of Di–Li–Liang (Ma et al., 2024).

4. Orthogonality-based extriangulated factorization systems and $$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$4-torsion pairs

A different but related notion is developed for extriangulated categories with negative first extensions. Here orthogonality is defined only for inflations. For inflations $$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$5 and $$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$6, one writes

$$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$7

if and only if

$$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$8

Given a class $$\mathfrak{s}(\delta)=[\,A \xrightarrow{i} B \xrightarrow{p} C\,],$$9 of inflations, the orthogonal classes are

$\mathbb{E}^{-1}$00

and

$\mathbb{E}^{-1}$01

An inflation factorization system is a pair $\mathbb{E}^{-1}$02 of classes of inflations such that every inflation factors as $\mathbb{E}^{-1}$03 with $\mathbb{E}^{-1}$04 and $\mathbb{E}^{-1}$05, and

$\mathbb{E}^{-1}$06

Deflation factorization systems are defined dually. In this usage, “extriangulated factorization systems” refers to either the inflation or the deflation version (Xu et al., 6 Jul 2025).

The corresponding object-theoretic notion is an $\mathbb{E}^{-1}$07-torsion pair $\mathbb{E}^{-1}$08, defined by

$\mathbb{E}^{-1}$09

These conditions yield, for every object $\mathbb{E}^{-1}$10, an $\mathbb{E}^{-1}$11-triangle

$\mathbb{E}^{-1}$12

with $\mathbb{E}^{-1}$13 and $\mathbb{E}^{-1}$14. In abelian categories, $\mathbb{E}^{-1}$15-torsion pairs coincide with classical torsion pairs; in triangulated categories they coincide with $\mathbb{E}^{-1}$16-structures.

The main theorem of Xu–Zhang–Zhu gives a bijection

$\mathbb{E}^{-1}$17

between $\mathbb{E}^{-1}$18-torsion pairs and inflation factorization systems. The dual bijection uses deflations and cocones. The factorization part is produced by taking an inflation $\mathbb{E}^{-1}$19 with $\mathbb{E}^{-1}$20-triangle

$\mathbb{E}^{-1}$21

decomposing $\mathbb{E}^{-1}$22 via an $\mathbb{E}^{-1}$23-triangle

$\mathbb{E}^{-1}$24

with $\mathbb{E}^{-1}$25 and $\mathbb{E}^{-1}$26, and then using $\mathbb{E}^{-1}$27 to obtain a diagram of conflations from which one reads off

$\mathbb{E}^{-1}$28

Orthogonality follows because cones of $\mathbb{E}^{-1}$29 lie in $\mathbb{E}^{-1}$30 and cones of $\mathbb{E}^{-1}$31 lie in $\mathbb{E}^{-1}$32, so both $\mathbb{E}^{-1}$33 and $\mathbb{E}^{-1}$34 vanish between them (Xu et al., 6 Jul 2025).

This theory does not assume a priori closure under composition or retracts, but these properties can be derived from the orthogonality axioms and the ET-axioms. In particular, if $\mathbb{E}^{-1}$35, then the cone of $\mathbb{E}^{-1}$36 lies in the extension-closed subcategory $\mathbb{E}^{-1}$37, so $\mathbb{E}^{-1}$38; retract stability is obtained by passing to retracts of cones (Xu et al., 6 Jul 2025).

5. Recollements and gluing

The recollement theory developed for $\mathbb{E}^{-1}$39-torsion pairs and inflation factorization systems requires a recollement $\mathbb{E}^{-1}$40 of extriangulated categories with the usual adjoint triples

$\mathbb{E}^{-1}$41

together with $\mathbb{E}^{-1}$42, full faithfulness of $\mathbb{E}^{-1}$43, and left and right exact $\mathbb{E}^{-1}$44-triangle sequences of the forms specified in conditions (R4) and (R5). The gluing results further assume balanced negative first extensions, exactness of $\mathbb{E}^{-1}$45 and $\mathbb{E}^{-1}$46, and that $\mathbb{E}^{-1}$47 preserves projectives (Xu et al., 6 Jul 2025).

Under these hypotheses, a key adjunction lemma identifies negative first extensions across the side categories: if $\mathbb{E}^{-1}$48 admits a left adjoint $\mathbb{E}^{-1}$49, the categories have balanced negative first extensions, $\mathbb{E}^{-1}$50 has enough projectives, and $\mathbb{E}^{-1}$51 is exact and preserves projectives, then

$\mathbb{E}^{-1}$52

The proof uses projective resolutions, exactness, and the Five-lemma.

If $\mathbb{E}^{-1}$53 is an $\mathbb{E}^{-1}$54-torsion pair in $\mathbb{E}^{-1}$55 and $\mathbb{E}^{-1}$56 is an $\mathbb{E}^{-1}$57-torsion pair in $\mathbb{E}^{-1}$58, their glued pair in $\mathbb{E}^{-1}$59 is defined by

$\mathbb{E}^{-1}$60

$\mathbb{E}^{-1}$61

The theorem states that $\mathbb{E}^{-1}$62 is again an $\mathbb{E}^{-1}$63-torsion pair. Orthogonality is verified by applying $\mathbb{E}^{-1}$64 and $\mathbb{E}^{-1}$65 to the recollement triangles and transporting vanishing through the adjunction lemma. The decomposition $\mathbb{E}^{-1}$66 is obtained by standard gluing arguments.

The factorization-system counterpart is defined by

$\mathbb{E}^{-1}$67

$\mathbb{E}^{-1}$68

where $\mathbb{E}^{-1}$69 and $\mathbb{E}^{-1}$70 are inflation factorization systems on the side categories. The glued pair $\mathbb{E}^{-1}$71 is an inflation factorization system on $\mathbb{E}^{-1}$72. The argument proceeds by first gluing the associated $\mathbb{E}^{-1}$73-torsion pairs and then recovering the factorization system using the main bijection; in particular, one checks

$\mathbb{E}^{-1}$74

The deflation case is entirely dual (Xu et al., 6 Jul 2025).

6. Specializations, relations, and further directions

The two theories specialize to well-known structures in exact, abelian, and triangulated settings. In an exact category, extriangles are short exact sequences, and for a cotorsion pair $\mathbb{E}^{-1}$75 the associated admissible weak factorization system is

$\mathbb{E}^{-1}$76

$\mathbb{E}^{-1}$77

In an abelian category, this recovers the classical correspondence with monomorphisms and epimorphisms. In a triangulated category, hereditary cotorsion pairs coincide with co-$\mathbb{E}^{-1}$78-structures, and the admissible weak factorization system criterion becomes a characterization of co-$\mathbb{E}^{-1}$79-structures by cancellation properties (Ma et al., 2024).

For the orthogonality-based theory, abelian categories identify inflations with monomorphisms and deflations with epimorphisms, and the bijections specialize to torsion pairs versus monomorphism or epimorphism factorization systems, unifying the Rosický–Tholen correspondence. In triangulated categories, every morphism is an inflation, $\mathbb{E}^{-1}$80-torsion pairs coincide with $\mathbb{E}^{-1}$81-structures, and the resulting factorization systems recover the Loregian–Virili correspondence between $\mathbb{E}^{-1}$82-structures and triangulated factorization systems (Xu et al., 6 Jul 2025).

The following comparison summarizes the two principal notions.

Aspect	Admissible weak factorization systems	Extriangulated factorization systems
Morphisms involved	All morphisms, with admissibility forcing inflations/deflations	Inflations or deflations only
Object-side correspondence	Cotorsion pairs	$\mathbb{E}^{-1}$83-torsion pairs
Orthogonality data	Lifting property $\mathbb{E}^{-1}$84 and $\mathbb{E}^{-1}$85	Vanishing of $\mathbb{E}^{-1}$86 and $\mathbb{E}^{-1}$87 between cones

Beyond exact and triangulated categories, the 2024 paper points to extriangulated categories arising as extension-closed subcategories of triangulated categories or from cluster-tilting settings, where cotorsion pairs induce admissible weak factorization systems and vice versa. The 2025 paper gives a concrete family in extended module categories $\mathbb{E}^{-1}$88, where an $\mathbb{E}^{-1}$89-term silting complex $\mathbb{E}^{-1}$90 determines an $\mathbb{E}^{-1}$91-torsion pair

$\mathbb{E}^{-1}$92

and hence an inflation factorization system $\mathbb{E}^{-1}$93 (Ma et al., 2024, Xu et al., 6 Jul 2025).

The relation to model structures is explicit on the admissible weak factorization system side. In extriangulated categories satisfying WIC, compatible hereditary cotorsion pairs can be used to build Hovey triples and model structures, and the AWFS–cotorsion-pair correspondence provides a route from cotorsion-theoretic data to admissible model structures once compatibility is verified. This suggests a structural parallel between weak factorization systems, cotorsion pairs, and model-categorical constructions in the extriangulated setting (Ma et al., 2024).

The papers also indicate several directions for further development. One is to explore admissible weak factorization systems beyond weakly idempotent complete settings and to identify minimal assumptions ensuring the cotorsion-pair bijection. Another is to extend the orthogonality-based factorization-system correspondence to admissible weak factorization systems, thereby relaxing the orthogonality regime. Additional directions include the study of $\mathbb{E}^{-1}$94-exangulated analogues, interactions with twin cotorsion pairs and hearts, and the weakening of gluing assumptions such as balanced negative extensions, exactness, or preservation of projectives (Ma et al., 2024, Xu et al., 6 Jul 2025).