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 $r \circ s = 1_B$.

$1 \to A \stackrel{i}{\to} B \stackrel{q}{\to} C \to 1$

is given by a section$j: C \to B$ of $q$. This yields an idempotent $\pi = j \circ q$; in the abelian category case, this yields a further idempotent $1_B - \pi$ which is canonically split by a further retraction of $i: A \to B$, thus yielding a biproduct diagram.

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

The splitting of an idempotent $e$ is both the limit and the colimit of the diagram containing only two parallelendomorphisms 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.

Examples

Proposition

Let $\mathcal{T}$ be a triangulated category such that in addition to the existence of small direct sums the objectwise direct sum of any small family of distinguished triangles is itself a distinguished triangle (Bökstedt-Neeman 93, def. 1.2).