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.Artin, 1972, p. 413. Recall that an idempotent morphism $p$ is an endomorphism of an object with the property that $p\circ p = p$ . Elementary considerations show that every idempotent then has a cokernel.Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A 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 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 $\operatorname{Kar}C$ together with a functor

: $s:C \to \operatorname{Kar}C$

such that the image $s(p)$ of every idempotent $p$ in $C$ splits in $\operatorname{Kar}C$ .

When applied to a preadditive category $C$ , the Karoubi envelope construction yields a pseudo-abelian category $\operatorname{Kar}C$

called the pseudo-abelian completion or pseudo-abelian envelope of $C$ . Moreover, the functor

: $C \to \operatorname{Kar}C$

is in fact an additive morphism.

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

: $f:(X,p) \to (Y,q)$

in $\operatorname{Kar}C$ are those morphisms

: $f:X \to Y$

such that $f = q \circ f = f \circ p$ in $C$ .

The functor

: $C \to \operatorname{Kar}C$

is given by taking $X$ to $(X, \mathrm{id}_X)$ .

Citations

References

{{cite book

| first = Michael

| last = Artin

| author-link = Michael Artin

|editor=Alexandre Grothendieck

|editor-link=Alexandre Grothendieck

|editor2=Jean-Louis Verdier

|editor2-link=Jean-Louis Verdier

| 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-Verlag

| location = Berlin; New York

| language = fr

| pages = xix+525

| no-pp = true

}}

Category:Category theory