is given by a section of . This yields an idempotent ; in the abelian category case, this yields a further idempotent which is canonically split by a further retraction of .
Any split idempotent is an idempotent, since
The splitting of an idempotent is both the limit and the colimit of the diagram containing only two parallel endomorphisms of , namely 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.
Let be a triangulated category such that in addition to the existence of direct sums (additivity) the objectwise direct sum of any two distinguished triangles is itself a distinguished triangles (Bökstedt-Neeman 93, def. 1.2).
Then in all idempotents split.