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preadditive category
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(Definition)
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A category
 is an ab-category or ab-category if
- for every pair of objects
 of
 , there is a binary operation called addition, written  or simply  , defined on
 ,
- the set
 , together with  is an abelian group,
- (left distributivity) if
 and
 , then
 ,
- (right distributivity) if
 and
 , then
 .
In a nutshell, an ab-category is a category in which every hom set in
 is an abelian group such that morphism composition distributes over addition. Ab in the name stands for abelian, clearly indicative of the second condition above.
Since a group has a multiplicative (or additive if abelian) identity,
 for every pair of objects  in
 . Furthermore, each  contains a unique morphism, written  , as the additive identity of  . Because the subset
of  is also a subgroup by right distributivity, and the additive identity of a subgroup coincides with the additive identity of the group, we have the following identity
There are many examples of ab-categories, including the category of abelian groups, the category of  -modules ( a ring), the category of chain complexes, and the category of rings (not necessarily containing a multiplicative identity). However, the category of rings with 1 is not an ab-category (see below for more detail). Nevertheless, a unital ring  itself considered as a category is an ab-category, as the ring of endomorphisms clearly forms an abelian group. It is in fact a ring! This can be seen as a special case of the fact that, in an ab-category,
 is always a ring (with 1). So, conversely, an ab-category with one object is a ring with 1, whose morphisms are elements of the ring.
If an ab-category has an initial object, that object is also a terminal object. By duality, the converse is also true. Therefore, in an ab-category, initial object, terminal object, and zero object are synonymous. In the category
 of unital rings,
 is an initial object, but it has no terminal object, therefore
 is not an ab-category.
An ab-category with a zero object  is called a preadditive category.
In a preadditive category, the groups  and  are trivial groups by the definition of the zero object  . Therefore, the zero morphism in  is also the additive identity of  :
Most of the examples of ab-categories are readily seen to be preadditive. If a preadditive category  has only one object, we see from above that it must be a ring. But this object must also be a zero object, so that
 must be trivial, which means  itself must be trivial too,  !
Remark. In some literature, a preadditive category is an ab-category, and some do not insist that a preadditive category contains a zero object. Here, we choose to differentiate the two.
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"preadditive category" is owned by CWoo.
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(view preamble)
Cross-references: differentiate, zero morphism, zero object, converse, duality, terminal object, initial object, ring of endomorphisms, unital ring, multiplicative identity, chain complexes, ring, subgroup, subset, contains, identity, additive, multiplicative, group, abelian, distributes over, composition, morphism, right distributivity, left distributivity, abelian group, addition, binary operation, objects, category
There are 4 references to this entry.
This is version 6 of preadditive category, born on 2006-05-16, modified 2006-05-23.
Object id is 7913, canonical name is PreadditiveCategory.
Accessed 1467 times total.
Classification:
| AMS MSC: | 18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories) |
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Pending Errata and Addenda
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