Idempotents

Idea

The notion of an idempotent morphism in a category generalizes the notion of projector in the context of linear algebra: it is an endomorphism $e \colon X \to X$ of some object $X$ that “squares to itself” in that the composition of $e$ with itself is again $e$:

Accordingly, given any idempotent $e \colon X \to X$ it is of interest to ask what subobject $A \stackrel{i}{\hookrightarrow} X$ of $X$ it is the projector onto, in that there is a projection $X \stackrel{p}{\to} A$ such that the idempotent is the composite of this projection followed by including $A$ back into $X$:

As opposed to the case of linear algebra, in general such a factorization into a projection onto a subobject $A$ need not actually exists for an idempotent $e$ in a generic category. If it exists, one says that $e$ is a split idempotent.

Accordingly, one is interested in those categories for which every idempotent is split. These are called idempotent complete categories or Cauchy complete categories. If a category is not yet idempotent complete it can be completed to one that is: its Karoubi envelope or Cauchy completion.

Definition

An endomorphism $e\colon B \to B$ in a category is an idempotent if the composition with itself equals itself

A splitting of an idempotent $e$ consists of morphisms $s\colon A \to B$ and $r\colon B \to A$ such that $r \circ s = 1_A$ and $s \circ r = e$. In this case $A$ is a retract of $B$, and we call $e$ a split idempotent.

Of course, we can simply consider the idempotent elements of any monoid.

Properties

The algebra of idempotents

Given an abelian monoid $R$, the idempotent elements form a submonoid $Idem(R)$.

Given a commutative ring $R$, the idempotent elements of $R$ form a Boolean algebra $Idem(R)$ with these operations:

• $\top \coloneqq 1$,
• $P \wedge Q \coloneqq P Q$,
• $\bot \coloneqq 0$,
• $P \vee Q \coloneqq P - P Q + Q$,
• $\neg{P} \coloneqq 1 - P$.

This is important in measure theory; if $R$ is the ring $L^\infty(X,\mathcal{M},\mathcal{N})$ of essentially bounded real-valued measurable functions on some measurable space $(X,\mathcal{M})$ modulo an ideal $\mathcal{N}$ of null sets, then $Idem(R)$ is the Boolean algebra of characteristic functions of measurable sets modulo null sets, which is isomorphic to the Boolean algebra $\mathcal{M}/\mathcal{N}$ of measurable sets modulo null sets itself.

If $R$ is a commutative $*$-ring, then we may restrict to the self-adjoint idempotent elements to get the Boolean algebra $Proj(R)$. In measure theory, if $R$ is the complex-valued version of $L^\infty(X,\mathcal{M},\mathcal{N})$, then $Proj(R)$ will still reconstruct $\mathcal{M}/\mathcal{N}$. In operator algebra theory, the self-adjoint idempotent elements of an operator algebra are called projection operators, which is the origin of the notation $Proj$. (Sometimes one requires projection operators to be proper: to have norm $1$; the only projection operator that is not proper is $0$.)

The projection operators of a commutative $W^\star$-algebra give the link between operator algebra theory and measure theory; in fact, the categories of commutative $W^\star$-algebras and of localisable measurable spaces (or measurable locales) are dual, and $W^\star$-algebra theory in general may be thought of as noncommutative measure theory. In noncommutative measure theory, the projection operators are still important, but they no longer form a Boolean algebra.

The universal idempotent-split completion

Given a category $\mathcal{C}$ one may ask for the universal category obtained from $\mathcal{C}$ subject to the constraint that all idempotents are turned into split idempotents. This is called the Karoubi envelope of $\mathcal{C}$. More generally, in enriched category theory it is called the Cauchy completion of $\mathcal{C}$.

Related concepts

• retract, section

• Cauchy complete category

• idempotent complete (infinity,1)-category

• idempotent semiring

References

• Saunders MacLane, §I.5 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

Formalization in homotopy type theory:

• Mike Shulman, Splitting idempotents