1 Background and Motivation

1.1 Idempotent morphisms and their significance

In an arbitrary category, an idempotent endomorphism is a morphism \(e:A\to A\) satisfying \(e\circ e=e\). Such maps behave like “projections” in many algebraic settings, because applying them twice has the same effect as applying them once. Even when the ambient category does not literally contain the “image” of \(e\) as a separate object, the idempotent already encodes a decomposition tendency: it singles out a part of \(A\) that is stable under the action of \(e\).

1.2 Splitting idempotents vs. retaining formal structure

A category is said to split idempotents if every idempotent endomorphism \(e\) can be realized via an actual direct summand: there exist an object \(B\) and morphisms that exhibit \(B\) as the “image” of \(e\). If the category does not split idempotents automatically, one often encounters constructions that are stable only “up to idempotent” rather than on the nose. Karoubi completion addresses this mismatch by changing the category so that these projection-like morphisms become genuine splittings, without losing the original categorical structure more than necessary.

1.3 Connection to direct summands and retracts

The categorical counterpart of “direct summands” is the notion of retracts. In an additive or more generally well-behaved setting, an idempotent determines a retract: an object \(B\) that sits inside \(A\) and is returned by a suitable retraction. Karoubi completion formalizes this principle in full generality: it builds a new category in which retracts corresponding to all idempotents are present as honest objects.

2 Definition of the Karoubi Completion

2.1 Objects as pairs (A, e) with an idempotent e

Given a category \(\mathcal{C}\), its Karoubi completion (also called the idempotent completion) \(\mathrm{Kar}(\mathcal{C})\) is defined by taking as objects pairs \((A,e)\), where \(A\) is an object of \(\mathcal{C}\) and \(e:A\to A\) is an idempotent endomorphism (\(e^2=e\)).

Intuitively, \((A,e)\) represents “the part of \(A\) cut out by \(e\).” If \(e\) is the identity on \(A\), then \((A,\mathrm{id}_A)\) recovers the original object in a tautological way.

2.2 Morphisms between split objects

For two objects \((A,e)\) and \((B,f)\) in \(\mathrm{Kar}(\mathcal{C})\), the morphisms are those morphisms \(u:A\to B\) in \(\mathcal{C}\) that respect the idempotents: \[ \mathrm{Hom}_{\mathrm{Kar}(\mathcal{C})}((A,e),(B,f)) =\{\,u:A\to B \mid u = f\circ u \circ e\,\}. \] This condition ensures that maps between “cut-out parts” factor correctly through the corresponding projection behavior encoded by \(e\) and \(f\).

2.3 Composition and identity in the completed category

Composition is inherited from \(\mathcal{C}\). If \(u:(A,e)\to(B,f)\) and \(v:(B,f)\to(C,g)\) satisfy \(u=fue\) and \(v=g vf\), then \(v\circ u\) automatically satisfies \[ g\circ (v\circ u)\circ e =(g\circ v\circ f)\circ u =v\circ (f\circ u)\;=\;v\circ u, \] so it is a valid morphism in \(\mathrm{Kar}(\mathcal{C})\). The identity morphism on \((A,e)\) is \(e:A\to A\), viewed as a morphism \((A,e)\to(A,e)\). Indeed, \(e\) satisfies \(e = e\circ e\circ e\), so it meets the defining constraint.

2.4 Relation to the category of retracts

Under mild standard hypotheses (for instance, in additive categories or in contexts where retracts are meaningful), objects \((A,e)\) correspond to retracts of \(A\). The idempotent itself provides the data of the inclusion and retraction after passing to the completion. In that sense, \(\mathrm{Kar}(\mathcal{C})\) is the “retract-closed” enlargement of \(\mathcal{C}\): every idempotent behaves as if its image were already an object.

3 Universal Property

3.1 The canonical functor into the completion

There is a canonical functor \(i:\mathcal{C}\to \mathrm{Kar}(\mathcal{C})\) defined by \[ i(A)=(A,\mathrm{id}_A), \qquad i(u)=u \] for a morphism \(u:A\to B\). The functor sends \(u\) to the same underlying morphism in \(\mathcal{C}\), and the idempotent compatibility is automatic because the relevant idempotents are identities.

3.2 Characterizing functors that preserve split idempotents

The Karoubi completion is characterized by a universal property: it is initial among functors from \(\mathcal{C}\) into categories in which all idempotents split. Concretely, if \(\mathcal{D}\) is a category and \(F:\mathcal{C}\to\mathcal{D}\) is a functor such that, for every idempotent \(e:A\to A\) in \(\mathcal{C}\), the morphism \(F(e)\) splits in \(\mathcal{D}\) (i.e., corresponds to an actual retract object), then there exists a functor \(\overline{F}:\mathrm{Kar}(\mathcal{C})\to\mathcal{D}\) with \(\overline{F}\circ i \cong F\).

At a conceptual level, \(\mathrm{Kar}(\mathcal{C})\) freely adjoins to \(\mathcal{C}\) an object representing each would-be image of an idempotent, along with the morphisms that make the representation compatible with composition.

3.3 Uniqueness up to equivalence

Because the universal property defines \(\mathrm{Kar}(\mathcal{C})\) uniquely up to equivalence, different realizations of “idempotent splitting” lead to equivalent completed categories. In practice, this means that while the construction depends on choosing objects \((A,e)\), the resulting enlargement is canonical in the sense that it does not depend on arbitrary auxiliary choices beyond categorical equivalence.

4 Properties and Basic Consequences

4.1 Idempotent completeness of the completion

By construction, \(\mathrm{Kar}(\mathcal{C})\) is idempotent complete: every idempotent in the completion splits as a retract. More strongly, the objects already encode idempotents, so splitting becomes built into the structure. This is one of the main reasons the Karoubi completion is used: it converts an “implicit” projection behavior into explicit summand objects.

4.2 Equivalences induced by different choices of splittings

In a category where an idempotent \(e\) splits, the resulting choice of summand data (an object and maps exhibiting a splitting) is typically not unique. However, any two such splittings are related by canonical isomorphisms or at least equivalences. The Karoubi completion packages these distinctions away by treating \((A,e)\) as the formal representative of the idempotent, so that different splitting choices are absorbed into the universal nature of the completion.

4.3 Interaction with additive and linear structures

When \(\mathcal{C}\) is additive (or enriched over abelian groups, vector spaces, etc.), the completion is compatible with that structure in the expected way. For example, morphism sets in \(\mathrm{Kar}(\mathcal{C})\) can inherit additive structure via the defining constraint \(u=fue\). Similarly, in linear or enriched categories, the completion can be formed so that enriched hom-objects remain well-defined and respect the enrichment while still enforcing idempotent splitting.

4.4 Behavior under categorical limits and colimits where applicable

Idempotent completion does not automatically preserve all limits and colimits in every generality, since it changes the object class. Nevertheless, it often behaves well with respect to constructions where idempotent splitting is compatible with the structure being formed. In algebraic and homological contexts, the completion is used as a “structural correction” that leaves many computations intact while making decompositions explicit.

5 Functoriality and Higher-Level Structure

5.1 Induced functors from ordinary functors

Given a functor \(F:\mathcal{C}\to\mathcal{D}\), there is an induced functor \(\mathrm{Kar}(F):\mathrm{Kar}(\mathcal{C})\to\mathrm{Kar}(\mathcal{D})\) defined by \[ \mathrm{Kar}(F)(A,e)=(F(A),F(e)), \] and on morphisms by applying \(F\) to the underlying morphism in \(\mathcal{C}\). Since \(F(e)\) remains idempotent, the assignment is well-defined. This makes Karoubi completion into a functorial process.

5.2 Compatibility with natural transformations

If \(\eta:F\Rightarrow G\) is a natural transformation between functors \(\mathcal{C}\to\mathcal{D}\), then it induces a natural transformation between the corresponding completed functors. The components at \((A,e)\) are constructed using the component maps of \(\eta\) at \(A\), and the idempotent constraints ensure that the induced maps respect the defining condition \(u=fue\). As a result, Karoubi completion is compatible with the 2-categorical viewpoint on functors and natural transformations.

5.3 Idempotent completion in enriched settings

In enriched settings, such as linear categories (enriched over vector spaces) or categories enriched over an abelian category, the completion can often be performed while keeping the enrichment intact. The key requirement is that the enrichment supplies morphism composition and identities in a way that interacts appropriately with idempotent equations. Under those circumstances, the idempotent-completing construction can be carried out “enrichedly,” preserving the ambient algebraic flavor of the hom-objects.

6 Examples

6.1 Trivial and minimal examples

If a category has only trivial idempotents (meaning every idempotent endomorphism is either an identity or behaves in a way that forces it to correspond to an existing object already), then the Karoubi completion may be essentially unchanged. In the extreme case of a category where endomorphisms admit no nontrivial idempotents, the completion adds no genuinely new objects.

6.2 Completion of a category with no nontrivial idempotents

Suppose \(\mathcal{C}\) is such that every idempotent \(e:A\to A\) is already equal to \(\mathrm{id}_A\). Then objects of \(\mathrm{Kar}(\mathcal{C})\) are of the form \((A,\mathrm{id}_A)\) only. Morphisms between \((A,\mathrm{id}_A)\) and \((B,\mathrm{id}_B)\) coincide with morphisms \(A\to B\) in \(\mathcal{C}\). Hence \(\mathrm{Kar}(\mathcal{C})\) is equivalent to \(\mathcal{C}\).

6.3 Examples involving matrix idempotents

In categories related to linear algebra, idempotents often appear as matrices \(e\) with \(e^2=e\). Such matrices correspond to projections onto subspaces. In a linear category that does not already include images as objects in the desired strict sense, Karoubi completion forces the existence of objects corresponding to these projection operators. The completed category then contains explicit representatives for these “projected” components, with morphisms constrained to respect the relevant projections.

6.4 Retracts in familiar algebraic categories

In module categories or other additive algebraic categories, retracts correspond to direct summands. Karoubi completion thus recovers the expected relationship between idempotents and summands: every idempotent gives rise to a direct summand object. When the original category already has this feature, the completion may not change the category up to equivalence; it formalizes and unifies the principle across broader categorical settings.

7 Relation to Other Constructions

7.1 Karoubi completion vs. Cauchy completion

The Cauchy completion is another enlargement procedure in which one freely adds objects to split certain kinds of “absolute” colimits or idempotent-like structures derived from enrichment. Karoubi completion can be viewed as the idempotent-centric aspect of such completions: where Cauchy completion often targets broader completeness notions, Karoubi completion specifically enforces splitting of idempotent endomorphisms.

7.2 Comparison with additive hulls and envelopes

An additive hull (or additive envelope) enlarges a category by formally adding finite biproducts and ensuring additive structure on hom-sets. Karoubi completion differs in focus: it does not primarily add sums or biproducts but rather ensures that idempotent decompositions are present as objects. In many settings, additive completion and idempotent completion can be combined, and the order can affect intermediate descriptions while leaving the eventual “stable” completion equivalent.

Although the Karoubi completion is a categorical construction rather than a ring-theoretic decomposition theorem, it resonates with ideas such as semisimplicity and semisplitting. Idempotents in algebraic structures often correspond to decomposition data, and forcing their splitting is closely aligned with how semisimple components emerge. The connection is typically formal: Karoubi completion provides the categorical environment in which decomposition-by-idempotent becomes literal.

8 Applications

8.1 Simplifying categorical decompositions

A recurring use of Karoubi completion is that it turns abstract decomposition patterns into concrete objects. This simplifies reasoning about morphisms that are naturally compatible with projections, because one can replace “maps factoring through an idempotent” by genuine morphisms between explicit summand objects in the completed category.

8.2 Use in homological algebra and stable phenomena (formal overview)

In homological algebra, stable phenomena such as direct-summand behavior and splitting of projector-like maps are common. When the ambient category lacks idempotent completeness, important constructions may only exist up to idempotent completion. Applying Karoubi completion provides a setting where projectors correspond to actual objects, which helps clarify the relationship between complexes, resolutions, and derived equivalences that depend on summand decompositions.

8.3 Structural cleanup in diagrammatic and representation-theoretic contexts

Diagrammatic approaches and representation theory frequently involve projector morphisms, “cutting out” subrepresentations, or passing to direct summands. Karoubi completion acts as a cleanup tool: it ensures that all such projector operations correspond to objects in the category. As a result, computations and conceptual arguments can be expressed without constantly appealing to implicit “image” objects that are not explicitly present.