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In [[mathematics]], specifically in [[category theory]], a '''pseudo-abelian category''' is a [[category (mathematics)|category]] that is [[preadditive category|preadditive]] and is such that every [[idempotent]] has a [[kernel (category theory)|kernel]]
.<ref>Artin, 1972, p. 413.</ref> Recall that an idempotent morphism <math>p</math> is an endomorphism of an object with the property that <math>p\circ p = p</math>. Elementary considerations show that every idempotent then has a cokernel.<ref>Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A</ref> 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 [[rng (algebra)|rngs]] (not [[ring (mathematics)|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>C</math> a category <math>kar(C)</math> together with a functor
:<math>s:C\rightarrow kar(C)</math>
such that the image <math>s(p)</math> of every idempotent <math>p</math> in <math>C</math> splits in <math>kar(C)</math>.
When applied to a [[preadditive category]] <math>C</math>, the Karoubi envelope construction yields a pseudo-abelian category <math>kar(C)</math>
called the pseudo-abelian completion of <math>C</math>. Moreover, the functor
:<math>C\rightarrow kar(C)</math>
is in fact an additive morphism.

To be precise, given a preadditive category <math>C</math> we construct a pseudo-abelian category <math>kar(C)</math> in the following way. The objects of <math>kar(C)</math> are pairs <math>(X,p)</math> where <math>X</math> is an object of <math>C</math> and <math>p</math> is an idempotent of <math>X</math>. The morphisms
:<math>f:(X,p)\rightarrow (Y,q)</math>
in <math>kar(C)</math> are those morphisms
:<math>f:X\rightarrow Y</math>
such that <math>f=q\circ f\circ p</math> in <math>C</math>.
The functor
:<math>C\rightarrow kar(C)</math>
is given by taking <math>X</math> to <math>(X,id_X)</math>.

== Citations ==

<references/>

== References ==

* {{cite book
 | first = Michael
 | last = Artin
 | authorlink = Michael Artin
 | coauthors = [[Alexandre Grothendieck]], [[Jean-Louis Verdier]], eds.
 | title = 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''')
 | year = 1972
 | publisher = [[Springer Science+Business Media|Springer-Verlag]]
 | location = Berlin; New York
 | language = French
 | pages = xix+525
 | nopp = true
}}

[[Category:Category theory]]