Pseudo-abelian category explained
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 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
functors:C\to\operatorname{Kar}C
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
.
References
- Book: Artin
, Michael
. Michael Artin . Alexandre Grothendieck . Alexandre Grothendieck . Jean-Louis Verdier . Jean-Louis Verdier . 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) . 1972 . . Berlin; New York . fr . xix+525 . true.
Notes and References
- Artin, 1972, p. 413.
- Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A
