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In
mathematics, an '''abelian category''' is a certain kind of
category in which
morphisms and objects can be added and in which
kernels and
cokernels exist and have the usual properties.
The motivating prototype example of an abelian category is the category of
abelian groups.
==Definitions==
It's possible to define abelian categories in a piecemeal fashion:
* A category is ''
preadditive'' if it is
enriched over the
monoidal category Ab of
abelian groups. This means that all morphism sets are abelian groups and the composition of morphisms is
bilinear.
* A preadditive category is ''
additive'' if every
finite set of objects has a
biproduct. This means that we can form finite
direct sums and
direct products.
* An additive category is ''
preabelian'' if every morphism has both a
kernel and a
cokernel.
* Finally, a preabelian category is ''abelian'' if every
monomorphism and every
epimorphism is
normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism.
It's also possible to define abelian categories all at once.
A category is ''abelian'' if it has:
*a
zero object;
*all
finitary limits;
*all finitary
colimits;
*only
normal monomorphisms;
*only normal
epimorphisms.
Most notably, this latter definition doesn't mention the enrichment over
Ab that began the piecemeal definition; that enrichment can be constructed from the assumptions above.
==Examples==
* As mentioned above, the category of all abelian groups is an abelian category. The category of all
finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.
* If
R is a
ring, then the category of all left (or right)
modules over
R is an abelian category. In fact, it can be shown that any abelian category is equivalent to a
full subcategory of such a category of modules (''
Mitchell's embedding theorem'').
* If
K is a
commutative noetherian ring, then the category of
finitely generated modules over
K is abelian. In this way, abelian categories show up in
commutative algebra.
* As special cases of the two previous examples: the category of
vector spaces over a fixed
field ''k'' is abelian, as is the category of finite-
dimensional vector spaces over ''k''.
* If
R is a ring, then the category of all
finitely presented left (or right) modules over
R is an abelian category. (The category of
finitely generated modules over
R is not always abelian.)
* If ''X'' is a
topological space, then the category of all
sheaves of abelian groups on ''X'' is an abelian category. More generally, the category of sheaves of abelian groups on a
Grothendieck site is an abelian category. In this way, abelian categories show up in
algebraic topology and
algebraic geometry.
* If
C is a
small category and
A is an abelian category, then the category of all
functors from
C to
A forms an abelian category (the morphisms of this category are the
natural transformations between functors). If
C is small and
preadditive, then the category of all
additive functors from
C to
A also forms an abelian category. The latter is a generalization of the ''R''-module example, since a ring can be understood as a preadditive category with a single object.
==Elementary properties==
Given any pair
A,
B of objects in an abelian category, there is a special
zero morphism from
A to
B.
This can be defined as the
zero element of the
hom-set Hom(
A,
B), since this is an abelian group.
Alternatively, it can be defined as the unique composition
A → 0 →
B, where 0 is the
zero object of the abelian category.
In an abelian category, every morphism
f can be written as the composition of an epimorphism followed by a monomorphism.
This epimorphism is called the ''
coimage'' of
f, while the monomorphism is called the ''
image'' of
f.
Subobjects and
quotient objects are
well-behaved in abelian categories.
For example, the
poset of subobjects of any given object
A is a
bounded lattice.
Every abelian category
A is a
module over the monoidal category of finitely generated abelian groups; that is, we can form a
tensor product of a finitely generated abelian group
G and any object
A of
A.
The abelian category is also a
comodule; Hom(
G,
A) can be interpreted as an object of
A.
If
A is
complete, then we can remove the requirement that
G be finitely generated; most generally, we can form finitary
enriched limits in
A.
==Related concepts==
Abelian categories are the most general setting for
homological algebra.
All of the constructions used in that field are relevant, such as
exact sequences, and especially
short exact sequences, and
derived functors.
Important theorems that apply in all abelian categories include the
five lemma (and the
short five lemma as a special case), as well as the
snake lemma (and the
nine lemma as a special case).
== History ==
Abelian categories were introduced by
Alexander Grothendieck in the middle of the
1950s in order to unify various
cohomology theories. At the time, there was a cohomology theory for
sheaves, and a cohomology theory for
groups. The two were defined completely differently, but they had formally almost identical properties. In fact, much of
category theory was developed as a language to study these similarities. Grothendieck managed to unify the two theories: they both arise as
derived functors on abelian categories; on the one hand the abelian category of sheaves of abelian groups on a topological space, on the other hand the abelian category of ''G''-modules for a given group ''G''.
==References==
* Barry Mitchell: ''Theory of Categories'', New York, Academic Press, 1965
* N. Popescu: ''Abelian categories with applications to rings and modules'', Academic Press, London, 1973
