nLab
split idempotent

Definition

A split idempotent in a category $C$ is a morphism $e:A\to A$ which has a retract, meaning an object $B$ and morphisms $r:A\to B$ and $s:B\to A$ such that $s\circ r=e$ but $r\circ s=1_B$.

Properties

• Any split idempotent is an idempotent, since

  $$e\circ e=(s\circ r)\circ (s\circ r)=s\circ (r\circ s)\circ r=s\circ 1_B\circ r=s\circ r=e.$$e \circ e = (s \circ r) \circ (s \circ r) = s \circ (r \circ s) \circ r = s \circ 1_B \circ r = s \circ r = e .

• The splitting of an idempotent $e$ is both the limit and the colimit of the diagram containing only two parallel endomorphisms of $A$, namely $e$ and the identity. Splittings of idempotents are preserved by any functor, making them absolute (co)limits. In ordinary (i.e. unenriched) categories, every absolute (co)limit can be constructed from split idempotents. Thus, the Cauchy completion of an ordinary (Set-enriched) category is just its completion under split idempotents.

• A category in which all idempotents split is called idempotent complete. The free completion of a category under split idempotents is also called its Karoubi envelope.