DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The idempotent completion of a preadditive category

Definition

Let $C$ be preadditive. Its idempotent completion $Kar (C)$ has:

objects: pairs $(A, e)$ with $A \in C$ and $e : A \to A$ idempotent;
morphisms $(A, e) \to (B, f)$ : morphisms $u : A \to B$ in $C$ satisfying $f u e = u$ ;
composition: inherited from $C$ ; and
identity on $(A, e)$ : the morphism $e$ itself.

The hom-set condition $f u e = u$ is equivalent to $f u = u = u e$ , so each hom-set is a subgroup of the ambient hom-group of Preadditive category.

The inherited composition is closed: if $u = f u e$ and $v = g v f$ , then $g (v u) e = g v f u e = v u .$ Moreover $f u = u = u e$ shows that the proposed identities $e$ and $f$ act as identities on every such $u$ . Thus the displayed data do form a category.

Depends on

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Dixy Msapato, The Karoubi envelope and weak idempotent completion of an extriangulated category, Definition 2.3 (standard reference, not scraped)