Becker Contraderived Category

• Becker contraderived category is a derived framework defined by homotopy-orthogonality to complexes of projectives in locally presentable abelian categories.
• It is realized through a projective abelian model structure, establishing a triangulated equivalence with the homotopy category of unbounded complexes of projectives.
• Extensions to DG-categories, CDG-modules, and contramodules highlight its versatility and impact in modern homological algebra.

The Becker contraderived category is a derived category of the second kind defined by homotopy-orthogonality to complexes of projective objects rather than by quotienting only by acyclic complexes. In the basic abelian setting, for a locally presentable abelian category $\mathsf B$ with a projective generator, a complex $X^\bullet\in K(\mathsf B)$ is called contraacyclic in the sense of Becker when

$$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$

and the Becker contraderived category is the Verdier quotient

$$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$

Positselski–Šťovíček showed that this category is realized by a projective abelian model structure and, under the stated hypotheses, is triangulated equivalent to the homotopy category of all unbounded complexes of projective objects, $K(\mathsf B_{\mathrm{proj}})$ (Positselski et al., 2021). Subsequent work extended the same pattern to abelian DG-categories, CDG-modules, contramodules over topological rings, and more general cotorsion-pair settings (Positselski et al., 2022, Positselski et al., 2024, Positselski et al., 2024, Positselski, 9 Sep 2025).

1. Becker’s definition and its ambient hypotheses

Becker’s formulation starts from an exact or abelian context with enough projectives and regards certain complexes as “invisible” to projectives. In the locally presentable abelian case treated by Positselski–Šťovíček, the relevant ambient category is a locally presentable abelian category $\mathsf B$ with enough projective objects; they show that this is the same as a locally presentable abelian category with a single projective generator $P$ (Positselski et al., 2021). A projective generator is a projective object such that every object of $\mathsf B$ is a quotient of a coproduct of copies of $P$.

With this hypothesis, the full thick subcategory $K(\mathsf B)_{\mathrm{ctr}}\subset K(\mathsf B)$ consists of the Becker-contraacyclic complexes, namely those orthogonal to every complex of projective objects. The corresponding quotient $X^\bullet\in K(\mathsf B)$0 is the contraderived category in the sense of Becker (Positselski et al., 2021). This definition differs from the ordinary derived category $X^\bullet\in K(\mathsf B)$1, because $X^\bullet\in K(\mathsf B)$2 is generally larger than the subcategory of acyclic complexes.

This framework is especially natural for locally presentable abelian categories that are not Grothendieck but still have enough projectives. The paper explicitly notes that many contramodule categories fall into this class and are often non-Grothendieck while remaining locally presentable with enough projectives (Positselski et al., 2021).

2. Projective model structure and identification with $X^\bullet\in K(\mathsf B)$3

The central structural theorem is the existence of a projective abelian model structure on the category $X^\bullet\in K(\mathsf B)$4 of unbounded complexes. Its Hovey triple is

$X^\bullet\in K(\mathsf B)$5

where $X^\bullet\in K(\mathsf B)$6 consists of all complexes of projective objects, $X^\bullet\in K(\mathsf B)$7 consists of the Becker-contraacyclic complexes, and $X^\bullet\in K(\mathsf B)$8 is the whole complex category (Positselski et al., 2021). Positselski–Šťovíček prove that this is a cofibrantly generated hereditary abelian model structure, called the contraderived model structure.

In this model structure, the cofibrant objects are exactly all complexes of projectives, the fibrant objects are all complexes, and the weak equivalences are the morphisms whose cones are contraacyclic (Positselski et al., 2021). Thus a map $X^\bullet\in K(\mathsf B)$9 is a weak equivalence precisely when $$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$0.

The resulting homotopy category identifies the Becker contraderived category with the homotopy category of projectives: $$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$1 More precisely, the composition

$$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$2

is a triangulated equivalence (Positselski et al., 2021). This is the characteristic Becker pattern: one does not restrict to homotopy projective complexes, but takes all complexes of projectives as cofibrant.

That distinction is essential. In the ordinary projective derived model structure on $$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$3, the cofibrant objects are the homotopy projective complexes of projectives, not all complexes of projectives. Positselski–Šťovíček show that these homotopy projectives are exactly the direct summands of transfinitely iterated extensions of the shifts $$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$4, i.e.

$$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$5

whereas the contraderived model structure has the strictly larger cofibrant class $$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$6 (Positselski et al., 2021).

3. Relation to coderived categories and to Positselski’s second-kind constructions

The coderived category is the injective-side dual. For a Grothendieck abelian category $$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$7, a complex $$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$8 is coacyclic in Becker’s sense when

$$\operatorname{Hom}_{K(\mathsf B)}(P^\bullet,X^\bullet)=0 \quad\text{for all }P^\bullet\in K(\mathsf B_{\mathrm{proj}}),$$9

and the corresponding model structure on $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$0 has all complexes cofibrant, complexes of injectives fibrant, and weak equivalences given by morphisms with coacyclic cones (Positselski et al., 2021). This duality is retained in later DG-abelian generalizations (Positselski et al., 2022).

A persistent conceptual issue is the comparison with Positselski’s older “coderived/contraderived categories of the second kind.” The papers distinguish two approaches. One is the elementary quotient approach, where the coacyclic or contraacyclic subcategory is the smallest triangulated subcategory containing totalizations of short exact sequences and closed under coproducts or products. The other is the model-categorical approach of Jørgensen, Krause, Becker, and others, where one builds model structures using cotorsion pairs and small object arguments (Positselski et al., 2021). Positselski–Šťovíček explicitly follow Becker’s approach in their contraderived and coderived constructions (Positselski et al., 2021, Positselski et al., 2022).

The comparison is only partially settled. The papers state that for DG-comodules over DG-coalgebras, and in several related settings, the two notions agree; but for general module categories it remains an open question whether Becker’s and Positselski’s contraderived or coderived categories coincide (Positselski et al., 2021, Positselski et al., 2022).

A broader unification appears in the cotorsion-pair framework. Given a hereditary complete cotorsion pair $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$1 generated by a set in a Grothendieck category $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$2, one has natural equivalences

$$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$3

so the Becker coderived category of the left-hand class and the Becker contraderived category of the right-hand class are both identified with the homotopy category of complexes over the core $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$4 (Positselski, 9 Sep 2025). This recasts Becker’s contraderived category as a “homotopy category of the core” in relative homological algebra.

4. Generation, well-generatedness, and size issues

The ordinary derived category $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$5 already exhibits strong generation properties in the locally presentable projective setting. If $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$6 is a projective generator of $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$7, then $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$8 is generated, as a triangulated category with coproducts, by the single object $$D_{\mathrm{ctr}}(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ctr}}.$$9 (Positselski et al., 2021). Equivalently, the full subcategory of homotopy projective complexes is the smallest triangulated subcategory containing $K(\mathsf B_{\mathrm{proj}})$0 and closed under coproducts.

On the contraderived side, the model structures are combinatorial and hereditary, and the resulting homotopy categories are well-generated triangulated categories (Positselski et al., 2021). In particular, $K(\mathsf B_{\mathrm{proj}})$1 is well-generated. The same pattern persists for the coderived category and, in the DG-abelian setting, for the Becker contraderived category of a locally presentable abelian DG-category (Positselski et al., 2022).

Set-theoretic control is supplied by the final section of (Positselski et al., 2021). There the authors introduce an exact category with an object size function and exact $K(\mathsf B_{\mathrm{proj}})$2-directed colimits of chains of admissible monomorphisms, and prove that the derived category of any such exact category has Hom sets. Since every locally presentable abelian category admits such a size function and exactness properties, it follows that the derived category of any locally presentable abelian category has Hom sets (Positselski et al., 2021). This is particularly relevant for quotient-style derived categories of the second kind, where large hom-classes could otherwise be a concern.

5. Abelian DG-categories and CDG-modules

The 2022 extension to abelian DG-categories places Becker’s contraderived category into a setting that simultaneously includes complexes in abelian categories, CDG-modules, and factorization categories (Positselski et al., 2022). For a locally presentable abelian DG-category $K(\mathsf B_{\mathrm{proj}})$3 with enough projectives, graded-projective objects are defined via projectivity in $K(\mathsf B_{\mathrm{proj}})$4, and an object $K(\mathsf B_{\mathrm{proj}})$5 is Becker-contraacyclic when

$K(\mathsf B_{\mathrm{proj}})$6

The resulting Verdier quotient

$K(\mathsf B_{\mathrm{proj}})$7

is the contraderived category in Becker’s sense (Positselski et al., 2022).

This DG-abelian version again admits a cofibrantly generated hereditary projective abelian model structure, now on $K(\mathsf B_{\mathrm{proj}})$8, with cofibrant objects the graded-projectives, fibrant objects all objects, and weakly trivial objects the Becker-contraacyclic ones (Positselski et al., 2022). The natural triangulated functor

$K(\mathsf B_{\mathrm{proj}})$9

is an equivalence, and the category is well-generated (Positselski et al., 2022).

For CDG-rings $\mathsf B$0, the 2024 paper on CDG-modules gives a particularly explicit family of equivalent models (Positselski et al., 2024). The Becker contraderived category of left CDG-modules is

$\mathsf B$1

where the projective model is the homotopy category of graded-projective CDG-modules (Positselski et al., 2024). It is also equivalent to the quotient

$\mathsf B$2

and to the homotopy category of graded-flat graded-cotorsion CDG-modules (Positselski et al., 2024). The paper further proves that the classes of cotorsion and graded-cotorsion CDG-modules coincide, and constructs two Quillen equivalent abelian model structures: a projective contraderived model structure and a flat contraderived model structure (Positselski et al., 2024).

Under the additional assumption that $\mathsf B$3 is graded right coherent, this CDG contraderived category is compactly generated, and its compact objects are anti-equivalent to the compact objects in the coderived category of right CDG-modules; these coderived compacts are identified with the idempotent completion of the absolute derived category of finitely presented right CDG-modules (Positselski et al., 2024).

6. Contramodules, compact generators, and later developments

A highly developed non-module example is the contraderived category of contramodules over a topological ring. For a complete, separated right linear topological ring $\mathsf B$4 with a countable base of neighborhoods of zero, the category $\mathsf B$5 is locally $\mathsf B$6-presentable and has an $\mathsf B$7-presentable projective generator (Positselski et al., 2024). Its Becker contraderived category is

$\mathsf B$8

and the natural functor

$\mathsf B$9

is a triangulated equivalence (Positselski et al., 2024).

Under the additional assumption that $P$0 is locally coherent, this contraderived category is compactly generated. Its compact objects are described by a contravariant fully faithful functor

$P$1

and one obtains a triangulated anti-equivalence

$P$2

(Positselski et al., 2024). The same paper proves a flat and projective periodicity theorem for contramodules and shows that, under the standing assumptions,

$P$3

directly paralleling Neeman’s module-theoretic picture (Positselski et al., 2024).

Later cotorsion-pair work shows that Becker co/contraderived categories are stable under passage between nested hereditary complete cotorsion pairs generated by sets, and that the induced comparison functors arise from Quillen adjunctions between abelian model structures on complexes (Positselski, 9 Sep 2025). In flat-type situations, the Becker coderived, ordinary derived, and Becker contraderived categories can all coincide; the criterion is expressed by relative periodicity conditions for the relevant cotorsion pairs (Positselski, 9 Sep 2025).

A neighboring development in deformation theory uses Positselski’s contraderived dg category rather than Becker’s, but the paper explicitly states that it provides a conceptual bridge between Positselski’s and Becker’s contraderived viewpoints (Pridham, 7 Mar 2025). There, Hinich’s simplicial nerve of the DGLA of derived derivations is shown to be equivalent to the space of deformations in the contraderived dg category, and when the underlying object is homologically bounded below, contraderived deformations are precisely derived deformations (Pridham, 7 Mar 2025). This suggests a broader role for contraderived categories of the second kind as the natural ambient categories for problems where ordinary derived categories are too small.

The Becker contraderived category therefore occupies a precise and robust place in contemporary homological algebra. In its classical form it is the Verdier quotient by projectively invisible complexes; in its model-categorical realization it is the homotopy category of all complexes of projectives; and in later generalizations it serves as a flexible triangulated framework for DG-abelian, curved, topological, and relative homological contexts (Positselski et al., 2021, Positselski et al., 2022, Positselski et al., 2024, Positselski et al., 2024, Positselski, 9 Sep 2025).