Pseudo-abelian category

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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

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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 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

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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

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  1. ^ Artin, 1972, p. 413.
  2. ^ Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A

References

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  • 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.