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 .[1] Recall that an idempotent morphism is an endomorphism of an object with the property that . Elementary considerations show that every idempotent then has a cokernel.[2] 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 a category together with a functor
such that the image of every idempotent in splits in . When applied to a preadditive category , the Karoubi envelope construction yields a pseudo-abelian category called the pseudo-abelian completion of . Moreover, the functor
is in fact an additive morphism.
To be precise, given a preadditive category we construct a pseudo-abelian category in the following way. The objects of are pairs where is an object of and is an idempotent of . The morphisms
in are those morphisms
such that in . The functor
is given by taking to .
Citations
References
- Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.). 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) (in French). Berlin; New York: Springer-Verlag. xix+525.
{{cite book}}
: Unknown parameter|nopp=
ignored (|no-pp=
suggested) (help)