Pseudo abelian category - Alchetron, the free social encyclopedia

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 p is an endomorphism of an object with the property that p ∘ p = p . 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.

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 C a category k a r ( C ) together with a functor

s : C → k a r ( C )

such that the image s ( p ) of every idempotent p in C splits in k a r ( C ) . When applied to a preadditive category C , the Karoubi envelope construction yields a pseudo-abelian category k a r ( C ) called the pseudo-abelian completion of C . Moreover, the functor

C → k a r ( C )

is in fact an additive morphism.

To be precise, given a preadditive category C we construct a pseudo-abelian category k a r ( C ) in the following way. The objects of k a r ( C ) are pairs ( X , p ) where X is an object of C and p is an idempotent of X . The morphisms

f : ( X , p ) → ( Y , q )

in k a r ( C ) are those morphisms

f : X → Y

such that f = q ∘ f = f ∘ p in C . The functor

C → k a r ( C )

is given by taking X to ( X , i d X ) .

References

Pseudo-abelian category Wikipedia

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Contents

Examples

Pseudo-abelian completion

References