The notion of an idempotent morphism in a category generalizes the notion of projector in the context of linear algebra: it is an endomorphism of some object that “squares to itself” in that the composition of with itself is again :
Accordingly, given any idempotent it is of interest to ask what subobject of it is the projector onto, in that there is a projection such that the idempotent is the composite of this projection followed by including back into :
As opposed to the case of linear algebra, in general such a factorization into a projection onto a subobject need not actually exists for an idempotent in a generic category. If it exists, one says that 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.
Of course, we can simply consider the idempotent elements of any monoid.
This is important in measure theory; if is the ring of essentially bounded real-valued measurable functions on some measurable space modulo an ideal of null sets, then is the Boolean algebra of characteristic functions of measurable sets modulo null sets, which is isomorphic to the Boolean algebra of measurable sets modulo null sets itself.
If is a commutative -ring, then we may restrict to the self-adjoint idempotent elements to get the Boolean algebra . In measure theory, if is the complex-valued version of , then will still reconstruct . In operator algebra theory, the self-adjoint idempotent elements of an operator algebra are called projection operators, which is the origin of the notation . (Sometimes one requires projection operators to be proper: to have norm ; the only projection operator that is not proper is .)
The projection operators of a commutative -algebra give the link between operator algebra theory and measure theory; in fact, the categories of commutative -algebras and of localisable measurable spaces (or measurable locales) are dual, and -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.
Given a category one may ask for the universal category obtained from subject to the constraint that all idempotents are turned into split idempotents. This is called the Karoubi envelope of . More generally, in enriched category theory it is called the Cauchy completion of .