category theory

# Contents

## 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 $r \circ s = 1_B$.

In an abelian or semi-abelian category, a splitting of an exact sequence

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

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 $i: A \to B$.

## Properties

• Any split idempotent is an idempotent, since

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

## Examples

###### Proposition

Let $\mathcal{T}$ 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 $\mathcal{T}$ all idempotents split.

## References

Revised on July 12, 2016 04:37:09 by Urs Schreiber (131.220.184.222)