Preadditive category

Preadditive category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear over the integers.

A preadditive category is also called an Ab-category, after the notation Ab for the category of abelian groups. Some authors have used the term additive category for preadditive categories, but Wikipedia follows the current trend of reserving this word for certain special preadditive categories (see special cases below).



The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. (Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism. In contrast, the category of all groups is not closed.) See medial category.

Other common examples:

  • The category of (left) modules over a ring R, in particular:
  • The algebra of matrices over a ring, thought of as a category as described in the article Additive category.
  • Any ring, thought of as a category with only one object, is a preadditive category. Here composition of morphisms is just ring multiplication and the unique hom-set is the underlying abelian group.

These will give you an idea of what to think of; for more examples, follow the links to special cases below.

Elementary properties

Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity of multiplication over addition.

Focusing on a single object A in a preadditive category, these facts say that the endomorphism hom-set Hom(A,A) is a ring, if we define multiplication in the ring to be composition. This ring is the endomorphism ring of A. Conversely, every ring (with identity) is the endomorphism ring of some object in some preadditive category. Indeed, given a ring R, we can define a preadditive category R to have a single object A, let Hom(A,A) be R, and let composition be ring multiplication. Since R is an Abelian group and multiplication in a ring is bilinear (distributive), this makes R a preadditive category. Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object (in the same way that a monoid can be viewed as a category with only one object - and forgetting the additive structure of the ring gives us a monoid).

In this way, preadditive categories can be seen as a generalisation of rings. Many concepts from ring theory, such as ideals, Jacobson radicals, and factor rings can be generalized in a straightforward manner to this setting. When attempting to write down these generalizations, one should think of the morphisms in the preadditive category as the "elements" of the "generalized ring". We won't go into such depth in this article.

Additive functors

If C and D are preadditive categories, then a functor FC → D is additive if it too is enriched over the category Ab. That is, F is additive if and only if, given any objects A and B of C, the function F: Hom(A,B) → Hom(F(A),F(B)) is a group homomorphism. Most functors studied between preadditive categories are additive.

For a simple example, if the rings R and S are represented by the one-object preadditive categories R and S, then a ring homomorphism from R to S is represented by an additive functor from R to S, and conversely.

If C and D are categories and D is preadditive, then the functor category Fun(C,D) is also preadditive, because natural transformations can be added in a natural way. If C is preadditive too, then the category Add(C,D) of additive functors and all natural transformations between them is also preadditive.

The latter example leads to a generalization of modules over rings: If C is a preadditive category, then Mod(C) := Add(C,Ab) is called the module category over C. When C is the one-object preadditive category corresponding to the ring R, this reduces to the ordinary category of (left) R-modules. Again, virtually all concepts from the theory of modules can be generalised to this setting.


Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the following biproduct condition:

The object B is a biproduct of the objects A1,...,An if and only if there are projection morphisms pjB → Aj and injection morphisms ijAj → B, such that (i1 o p1) + ··· + (in o pn) is the identity morphism of B, pj o ij is the identity morphism of Aj, and pj o ik is the zero morphism from Ak to Aj whenever j and k are distinct.

This biproduct is often written A1 ⊕ ··· ⊕ An, borrowing the notation for the direct sum. This is because the biproduct in well known preadditive categories like Ab is the direct sum. However, although infinite direct sums make sense in some categories, like Ab, infinite biproducts do not make sense.

The biproduct condition in the case n = 0 simplifies drastically; B is a nullary biproduct if and only if the identity morphism of B is the zero morphism from B to itself, or equivalently if the hom-set Hom(B,B) is the trivial ring. Note that because a nullary biproduct will be both terminal (a nullary product) and coterminal (a nullary coproduct), it will in fact be a zero object. Indeed, the term "zero object" originated in the study of preadditive categories like Ab, where the zero object is the zero group.

A preadditive category in which every biproduct exists (including a zero object) is called additive. Further facts about biproducts that are mainly useful in the context of additive categories may be found under that subject.

Kernels and cokernels

Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. That is, if fA → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism. Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category.

When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of fA → B with its embedding K → A. However, in a general preadditive category there may exist morphisms without kernels and/or cokernels.

There is a convenient relationship between the kernel and cokernel and the Abelian group structure on the hom-sets. Given parallel morphisms f and g, the equaliser of f and g is just the kernel of g − f, if either exists, and the analogous fact is true for coequalisers. The alternative term "difference kernel" for binary equalisers derives from this fact.

A preadditive category in which all biproducts, kernels, and cokernels exist is called pre-Abelian. Further facts about kernels and cokernels in preadditive categories that are mainly useful in the context of pre-Abelian categories may be found under that subject.

Special cases

Most of these special cases of preadditive categories have all been mentioned above, but they're gathered here for reference.

The preadditive categories most commonly studied are in fact Abelian categories; for example, Ab is an Abelian category.


  • Nicolae Popescu; 1973; Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Category of rings — In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity). Like many categories in mathematics, the category of rings is… …   Wikipedia

  • Category of abelian groups — In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category.The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the… …   Wikipedia

  • Category (mathematics) — In mathematics, a category is an algebraic structure that comprises objects that are linked by arrows . A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A …   Wikipedia

  • Category of topological spaces — In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again… …   Wikipedia

  • Category of sets — In mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics.Properties of the category of setsThe… …   Wikipedia

  • Additive category — In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A 1,..., A n of C have a biproduct A 1 ⊕ ⋯ ⊕ A n in C. (Recall that a category C is preadditive if all its… …   Wikipedia

  • Abelian category — In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of… …   Wikipedia

  • Functor category — In category theory, a branch of mathematics, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. Functor categories are of… …   Wikipedia

  • Kernel (category theory) — In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel… …   Wikipedia

  • Outline of category theory — The following outline is provided as an overview of and guide to category theory: Category theory – area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”