nLab
idempotent

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 $e:X\to X$ of some object $X$ that “squares to itself” in that the composition of $e$ with itself is again $e$:

$$e\circ e\simeq e.$$e \circ e \simeq e \,.

Accordingly, given any idempotent $e:X\to X$ 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 $X$:

$$e:X\stackrel{p}{\to }A\stackrel{i}{\hookrightarrow }X.$$e \colon X \stackrel{p}{\to} A \stackrel{i}{\hookrightarrow} 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 are Cauchy complete categories. If a category is not yet idempotent complete it can be completed to one that is: its Karoubi envelope or Cauchy completions.

Definition

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

$$e\circ e=e.$$e \circ e = e \,.

A splitting of an idempotent $e$ consists of morphisms $s:A\to B$ and $r:B\to A$ 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? $\mathrm{Idem}(R)$.

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

• $\top ≔1$,
• $P\wedge Q≔PQ$,
• $\perp ≔0$,
• $P\vee Q≔P-PQ+Q$,
• $\neg P≔1-P$.

This is important in measure theory; if $R$ is the ring $L^{\mathrm{\infty }}(X,ℳ,𝒩)$ of essentially bounded? real-valued measurable functions on some measurable space $(X,ℳ)$ modulo an ideal $𝒩$ of null sets, then $\mathrm{Idem}(R)$ is the Boolean algebra of characteristic functions of measurable sets modulo null sets, which is isomorphic to the Boolean algebra $ℳ/𝒩$ 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 $\mathrm{Proj}(R)$. In measure theory, if $R$ is the complex-valued version of $L^{\mathrm{\infty }}(X,ℳ,𝒩)$, then $\mathrm{Proj}(R)$ will still reconstruct $ℳ/𝒩$. In operator algebra theory, the self-adjoint idempotent elements of an operator algebra are called projection operators, which is the origin of the notation $\mathrm{Proj}$. (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 $𝒞$ one may ask for the universal category obtained from $𝒞$ subject to the constraint that all idempotents are turned into split idempotents. This is called the Karoubi envelope of $𝒞$. More generally, in enriched category theory it is called the Cauchy completion of $𝒞$.

Related concepts

• retract, section

• Cauchy complete category

• idempotent complete (infinity,1)-category

• idempotent semiring