nLab split idempotent

Definition

A split idempotent in a category $C$ is a morphism $e: A \to A$ which has a retract, meaning there exists 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 $i: A \to B$, thus yielding a biproduct diagram.

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 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).

Then in $\mathcal{T}$ all idempotents split.

(Bökstedt-Neeman 93, prop. 3.2)

Remark

Prop. is false under the weaker hypothesis of only binary/finite direct sums. A counter-example is in Schnürer 11, Remark 3.2

Related concepts

• Cauchy complete category

• idempotent completion

• pseudo-abelian category

References

• Francis Borceux, Section 6.5 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]

In triangulated categories:

• Marcel Bökstedt, Amnon Neeman, Homotopy limits in triangulated categories, Compositio Mathematica, tome 86, no 2 (1993) p. 209-234 (numdam)

• Olaf M. Schnürer, Homotopy categories and idempotent completeness, weight structures and weight complex functors (arXiv:1107.1227)

On split idempotents in Kleisli categories:

• Rene Guitart and J. Riguet. Enveloppe Karoubienne de catégories de Kleisli, Cahiers de topologie et géometrie différentielle catégoriques 33.3 (1992): 261-266. (Numdam)

• Robert Rosebrugh and Richard Wood, Split structures, TAC 13.12 (2004): 172-183. (TAC)

• Matías Menni, Algebraic categories whose projectives are explicitly free , TAC 22 no.20 (2009) pp.509-541. (abstract)

• Dusko Pavlovic and Peter-Michael Seidel. Quotients in monadic programming: Projective algebras are equivalent to coalgebras, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, 2017.