nLab
idempotent

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:XXe \colon X \to X of some object XX that “squares to itself” in that the composition of ee with itself is again ee:

eee. e \circ e \simeq e \,.

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

e:XpAiX. 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 AA need not actually exists for an idempotent ee in a generic category. If it exists, one says that ee 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:BBe\colon B \to B in a category is an idempotent if the composition with itself equals itself

ee=e. e \circ e = e \,.

A splitting of an idempotent ee consists of morphisms s:ABs\colon A \to B and r:BAr\colon B \to A such that rs=1 Ar \circ s = 1_A and sr=es \circ r = e. In this case AA is a retract of BB, and we call ee a split idempotent.

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

Properties

The algebra of idempotents

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

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

  • 1\top \coloneqq 1,
  • PQPQP \wedge Q \coloneqq P Q,
  • 0\bot \coloneqq 0,
  • PQPPQ+QP \vee Q \coloneqq P - P Q + Q,
  • ¬P1P\neg{P} \coloneqq 1 - P.

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

If RR is a commutative **-ring, then we may restrict to the self-adjoint idempotent elements to get the Boolean algebra Proj(R)Proj(R). In measure theory, if RR is the complex-valued version of L (X,,𝒩)L^\infty(X,\mathcal{M},\mathcal{N}), then Proj(R)Proj(R) will still reconstruct /𝒩\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 ProjProj. (Sometimes one requires projection operators to be proper: to have norm 11; the only projection operator that is not proper is 00.)

The projection operators of a commutative W W^\star-algebra give the link between operator algebra theory and measure theory; in fact, the categories of commutative W W^\star-algebras and of localisable measurable spaces (or measurable locales) are dual, and W 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\mathcal{C} one may ask for the universal category obtained from 𝒞\mathcal{C} subject to the constraint that all idempotents are turned into split idempotents. This is called the Karoubi envelope of 𝒞\mathcal{C}. More generally, in enriched category theory it is called the Cauchy completion of 𝒞\mathcal{C}.

Revised on November 2, 2012 12:32:59 by Urs Schreiber (82.169.65.155)