(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
Let be a category and an idempotent endomorphism of an object . One says that admits an image if the functor is representable, and the representing object is called the image of . Here is the functor mapping
in other words the image of is the difference kernel of , when it exists.
Now is called Karoubian if every idempotent admits an image. Since is idempotent iff is idempotent, this is the same as saying every idempotent has a kernel.
One can show that for any idempotent , is representable if and only if is, and that in fact their representing objects are canonically isomorphic.
Recall that one says splits if there exists an object , and morphisms , , such that and . Observe that when admits an image , it splits: by definition there are functorial isomorphisms for all between the image of the functor and ; now take the morphism corresponding to via , the morphism corresponding to via . Conversely, if splits via a pair , then is a difference kernel of : we have , and if satisfies , then clearly factors through , and uniquely so since sections are monomorphisms.
There is a universal functor from the category of (say, small) preadditive categories to the category of Karoubian categories, the Karoubinization functor; its value on a preadditive category is also called the Karoubian envelope or the pseudo-abelian completion of .
In more detail, there exists a Karoubian category associated to any category , and a fully faithful functor , which is universal in the sense that for any Karoubian category , the functor
taking a functor to the composite is an equivalence of categories. is called the Karoubi envelope of (aka the Cauchy completion, or the idempotent-splitting completion). It can be realized explicitly by taking as objects pairs , with idempotent, and as morphisms the morphisms that satisfy .