---
title: Becker Contraderived Category
url: https://www.emergentmind.com/topics/becker-contraderived-category
type: topic
---

# Becker Contraderived Category

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, \(K(\mathsf B_{\mathrm{proj}})\) [2101.10797]. Subsequent work extended the same pattern to abelian DG-categories, CDG-modules, contramodules over topological rings, and more general cotorsion-pair settings [2210.08237] [2401.07021] [2412.20494] [2509.07645].

## 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 \(P\) [2101.10797]. 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 \(D_{\mathrm{ctr}}(\mathsf B)\) is the contraderived category in the sense of Becker [2101.10797]. This definition differs from the ordinary derived category \(D(\mathsf B)=K(\mathsf B)/K(\mathsf B)_{\mathrm{ac}}\), because \(K(\mathsf B)_{\mathrm{ctr}}\) 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 [2101.10797].

## 2. Projective model structure and identification with \(K(\mathsf B_{\mathrm{proj}})\)

The central structural theorem is the existence of a projective abelian model structure on the category \(C(\mathsf B)\) of unbounded complexes. Its Hovey triple is
\[
\bigl(\mathcal L,\mathcal W,\mathcal R\bigr)
=
\bigl(C(\mathsf B_{\mathrm{proj}}),\, C(\mathsf B)_{\mathrm{ctr}},\, C(\mathsf B)\bigr),
\]
where \(\mathcal L\) consists of all complexes of projective objects, \(\mathcal W\) consists of the Becker-contraacyclic complexes, and \(\mathcal R\) is the whole complex category [2101.10797]. 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 [2101.10797]. Thus a map \(f:X^\bullet\to Y^\bullet\) is a weak equivalence precisely when \(\mathrm{cone}(f)\in C(\mathsf B)_{\mathrm{ctr}}\).

The resulting homotopy category identifies the Becker contraderived category with the homotopy category of projectives:
\[
\operatorname{Ho}\bigl(C(\mathsf B)_{\text{contraderived model}}\bigr)
\cong
D_{\mathrm{ctr}}(\mathsf B)
\cong
K(\mathsf B_{\mathrm{proj}}).
\]
More precisely, the composition
\[
K(\mathsf B_{\mathrm{proj}})\longrightarrow K(\mathsf B)\longrightarrow D_{\mathrm{ctr}}(\mathsf B)
\]
is a triangulated equivalence [2101.10797]. 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 \(C(\mathsf B)\), 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 \(P[i]\), i.e.
\[
C(\mathsf B_{\mathrm{proj}})^{\mathrm{hpr}}=\mathrm{Fil}(S)^\oplus,
\qquad S=\{P[i]\}_{i\in\mathbb Z},
\]
whereas the contraderived model structure has the strictly larger cofibrant class \(C(\mathsf B_{\mathrm{proj}})\) [2101.10797].

## 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 \(\mathsf A\), a complex \(X^\bullet\in K(\mathsf A)\) is coacyclic in Becker’s sense when
\[
\operatorname{Hom}_{K(\mathsf A)}(X^\bullet,J^\bullet)=0
\quad\text{for all }J^\bullet\in K(\mathsf A_{\mathrm{inj}}),
\]
and the corresponding model structure on \(C(\mathsf A)\) has all complexes cofibrant, complexes of injectives fibrant, and weak equivalences given by morphisms with coacyclic cones [2101.10797]. This duality is retained in later DG-abelian generalizations [2210.08237].

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 [2101.10797]. Positselski–Šťovíček explicitly follow Becker’s approach in their contraderived and coderived constructions [2101.10797] [2210.08237].

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 [2101.10797] [2210.08237].

A broader unification appears in the cotorsion-pair framework. Given a hereditary complete cotorsion pair \((\mathsf A,\mathsf B)\) generated by a set in a Grothendieck category \(\mathsf K\), one has natural equivalences
\[
\mathsf D^{\mathsf bco}(\mathsf A)\simeq \mathsf{Hot}(\mathsf A\cap\mathsf B)\simeq \mathsf D^{\mathsf bctr}(\mathsf B),
\]
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 \(\mathsf A\cap\mathsf B\) [2509.07645]. 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 \(D(\mathsf B)\) already exhibits strong generation properties in the locally presentable projective setting. If \(P\) is a projective generator of \(\mathsf B\), then \(D(\mathsf B)\) is generated, as a triangulated category with coproducts, by the single object \(P\) [2101.10797]. Equivalently, the full subcategory of homotopy projective complexes is the smallest triangulated subcategory containing \(P\) 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 [2101.10797]. In particular, \(D_{\mathrm{ctr}}(\mathsf B)\cong K(\mathsf B_{\mathrm{proj}})\) 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 [2210.08237].

Set-theoretic control is supplied by the final section of [2101.10797]. There the authors introduce an exact category with an object size function and exact \(\kappa\)-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 [2101.10797]. 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 [2210.08237]. For a locally presentable abelian DG-category \(\mathbf B\) with enough projectives, graded-projective objects are defined via projectivity in \(Z^0(\mathbf B^\natural)\), and an object \(X\in\mathbf B\) is Becker-contraacyclic when
\[
\operatorname{Hom}_{H^0(\mathbf B)}(Q,X)=0
\quad\text{for all graded-projective }Q\in \mathbf B_{\mathrm{proj}}.
\]
The resulting Verdier quotient
\[
D^{\mathrm{bctr}}(\mathbf B)=H^0(\mathbf B)/H^0(\mathbf B)_{\mathrm{ac}^{\mathrm{bctr}}}
\]
is the contraderived category in Becker’s sense [2210.08237].

This DG-abelian version again admits a cofibrantly generated hereditary projective abelian model structure, now on \(Z^0(\mathbf B)\), with cofibrant objects the graded-projectives, fibrant objects all objects, and weakly trivial objects the Becker-contraacyclic ones [2210.08237]. The natural triangulated functor
\[
H^0(\mathbf B_{\mathrm{proj}})\longrightarrow D^{\mathrm{bctr}}(\mathbf B)
\]
is an equivalence, and the category is well-generated [2210.08237].

For CDG-rings \(B^\bullet=(B^*,d,h)\), the 2024 paper on CDG-modules gives a particularly explicit family of equivalent models [2401.07021]. The Becker contraderived category of left CDG-modules is
\[
D^{\mathrm{bctr}}(B\text{-Mod})
=
H^0(B\text{-Mod}_{\mathrm{proj}})
\simeq
H^0(B\text{-Mod})/\mathsf{Ac}^{\mathrm{ctr}}(B\text{-Mod}),
\]
where the projective model is the homotopy category of graded-projective CDG-modules [2401.07021]. It is also equivalent to the quotient
\[
H^0(B\text{-Mod}_{\mathrm{flat}})\big/H^0(B\text{-Mod})_{\mathrm{flat}},
\]
and to the homotopy category of graded-flat graded-cotorsion CDG-modules [2401.07021]. 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 [2401.07021].

Under the additional assumption that \(B^*\) 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 [2401.07021].

## 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 \(\mathfrak R\) with a countable base of neighborhoods of zero, the category \(\mathfrak R\text{-Contra}\) is locally \(\aleph_1\)-presentable and has an \(\aleph_1\)-presentable projective generator [2412.20494]. Its Becker contraderived category is
\[
\mathbf D^{\mathrm{ctr}}(\mathfrak R\text{-Contra})
=
\mathrm{Hot}(\mathfrak R\text{-Contra})\big/\mathrm{Acb}^{\mathrm{ctr}}(\mathfrak R\text{-Contra}),
\]
and the natural functor
\[
\mathrm{Hot}(\mathfrak R\text{-Contraproj})\longrightarrow
\mathbf D^{\mathrm{ctr}}(\mathfrak R\text{-Contra})
\]
is a triangulated equivalence [2412.20494].

Under the additional assumption that \(\mathrm{Discr}\text{-}\mathfrak R\) is locally coherent, this contraderived category is compactly generated. Its compact objects are described by a contravariant fully faithful functor
\[
E:\mathbf D^{\mathrm b}(\mathrm{coh}\text{-}\mathfrak R)^{\mathrm{op}}
\longrightarrow
\mathbf D^{\mathrm{ctr}}(\mathfrak R\text{-Contra}),
\]
and one obtains a triangulated anti-equivalence
\[
\mathbf D^{\mathrm b}(\mathrm{coh}\text{-}\mathfrak R)
\simeq
\bigl(\mathbf D^{\mathrm{ctr}}(\mathfrak R\text{-Contra})^{\mathrm c}\bigr)^{\mathrm{op}}
\]
[2412.20494]. The same paper proves a flat and projective periodicity theorem for contramodules and shows that, under the standing assumptions,
\[
\mathrm{Hot}(\mathfrak R\text{-Contraproj})
\simeq
\mathbf D(\mathfrak R\text{-Contraflat})
\simeq
\mathbf D^{\mathrm{ctr}}(\mathfrak R\text{-Contra}),
\]
directly paralleling Neeman’s module-theoretic picture [2412.20494].

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 [2509.07645]. 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 [2509.07645].

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 [2503.05317]. 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 [2503.05317]. 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 [2101.10797] [2210.08237] [2401.07021] [2412.20494] [2509.07645].

Source: https://www.emergentmind.com/topics/becker-contraderived-category