split idempotent




[[!include idempotents - contents]]

Category theory

[[!include category theory - contents]]



A split idempotent in a category CC is a morphism e:AAe: A \to A which has a retract, meaning an object BB and morphisms r:ABr: A \to B and s:BAs: B \to A such that sr=es \circ r = e but rs=1 Br \circ s = 1_B.

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

1AiBqC11 \to A \stackrel{i}{\to} B \stackrel{q}{\to} C \to 1

is given by a section j:CBj: C \to B of qq. This yields an idempotent π=jq\pi = j \circ q; in the abelian category case, this yields a further idempotent 1 Bπ1_B - \pi which is canonically split by a further retraction of i:ABi: A \to B.


  • Any split idempotent is an idempotent, since

    ee=(sr)(sr)=s(rs)r=s1 Br=sr=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 ee is both the limit and the colimit of the diagram containing only two parallel endomorphisms of AA, namely ee 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.



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

(Bökstedt-Neeman 93, prop. 3.2)


In triangulated categories:

Last revised on July 12, 2016 at 04:37:09. See the history of this page for a list of all contributions to it.