Pseudo-abelian category

In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel. Recall that an idempotent morphism [math]\displaystyle{ p }[/math] is an endomorphism of an object with the property that [math]\displaystyle{ p\circ p = p }[/math]. Elementary considerations show that every idempotent then has a cokernel. The pseudo-abelian condition is stronger than preadditivity, but it is weaker than the requirement that every morphism have a kernel and cokernel, as is true for abelian categories. Synonyms in the literature for pseudo-abelian include pseudoabelian and Karoubian.

Examples

Any abelian category, in particular the category Ab of abelian groups, is pseudo-abelian. Indeed, in an abelian category, every morphism has a kernel.

The category of associative rngs (not rings!) together with multiplicative morphisms is pseudo-abelian.

A more complicated example is the category of Chow motives. The construction of Chow motives uses the pseudo-abelian completion described below.

Pseudo-abelian completion

The Karoubi envelope construction associates to an arbitrary category [math]\displaystyle{ C }[/math] a category [math]\displaystyle{ kar(C) }[/math] together with a functor

[math]\displaystyle{ s:C\rightarrow kar(C) }[/math]

such that the image [math]\displaystyle{ s(p) }[/math] of every idempotent [math]\displaystyle{ p }[/math] in [math]\displaystyle{ C }[/math] splits in [math]\displaystyle{ kar(C) }[/math]. When applied to a preadditive category [math]\displaystyle{ C }[/math], the Karoubi envelope construction yields a pseudo-abelian category [math]\displaystyle{ kar(C) }[/math] called the pseudo-abelian completion of [math]\displaystyle{ C }[/math]. Moreover, the functor

[math]\displaystyle{ C\rightarrow kar(C) }[/math]

is in fact an additive morphism.

To be precise, given a preadditive category [math]\displaystyle{ C }[/math] we construct a pseudo-abelian category [math]\displaystyle{ kar(C) }[/math] in the following way. The objects of [math]\displaystyle{ kar(C) }[/math] are pairs [math]\displaystyle{ (X,p) }[/math] where [math]\displaystyle{ X }[/math] is an object of [math]\displaystyle{ C }[/math] and [math]\displaystyle{ p }[/math] is an idempotent of [math]\displaystyle{ X }[/math]. The morphisms

[math]\displaystyle{ f:(X,p)\rightarrow (Y,q) }[/math]

in [math]\displaystyle{ kar(C) }[/math] are those morphisms

[math]\displaystyle{ f:X\rightarrow Y }[/math]

such that [math]\displaystyle{ f=q\circ f = f \circ p }[/math] in [math]\displaystyle{ C }[/math]. The functor

[math]\displaystyle{ C\rightarrow kar(C) }[/math]

is given by taking [math]\displaystyle{ X }[/math] to [math]\displaystyle{ (X,id_X) }[/math].

References

• Artin, Michael (1972). Alexandre Grothendieck. ed (in fr). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269). Berlin; New York: Springer-Verlag. pp. xix+525.