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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically in
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, an additive category is a preadditive category C admitting all
finitary In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values. In standard mathematics, an operat ...
biproducts.


Definition

There are two equivalent definitions of an additive category: One as a category equipped with additional structure, and another as a category equipped with ''no extra structure'' but whose objects and morphisms satisfy certain equations.


Via preadditive categories

A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the
monoidal category In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (cate ...
of abelian groups. In a preadditive category, every finitary product is necessarily a coproduct, and hence a biproduct, and conversely every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). The empty product, is a final object and the empty product in the case of an empty diagram, an
initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...
. Both being limits, they are not finite products nor coproducts. Thus an additive category is equivalently described as a preadditive category admitting all finitary products and with the null object or a preadditive category admitting all finitary coproducts and with the null object


Via semiadditive categories

We give an alternative definition. Define a semiadditive category to be a category (note: not a preadditive category) which admits a zero object and all binary biproducts. It is then a remarkable theorem that the Hom sets naturally admit an abelian monoid structure. A proof of this fact is given below. An additive category may then be defined as a semiadditive category in which every morphism has an
additive inverse In mathematics, the additive inverse of an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero el ...
. This then gives the Hom sets an abelian group structure instead of merely an abelian monoid structure.


Generalization

More generally, one also considers additive -linear categories for a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
. These are categories enriched over the monoidal category of - modules and admitting all finitary biproducts.


Examples

The original example of an additive category is the
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: indeed, every small abelian category can be embedded in Ab. Properties The zero object o ...
Ab. The zero object is the trivial group, the addition of morphisms is given pointwise, and biproducts are given by direct sums. More generally, every module category over a ring is additive, and so in particular, the category of vector spaces over a field is additive. The algebra of matrices over a ring, thought of as a category as described below, is also additive.


Internal characterisation of the addition law

Let C be a semiadditive category, so a category having all finitary biproducts. Then every hom-set has an addition, endowing it with the structure of an abelian monoid, and such that the composition of morphisms is bilinear. Moreover, if C is additive, then the two additions on hom-sets must agree. In particular, a semiadditive category is additive
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
every morphism has an additive inverse. This shows that the addition law for an additive category is ''internal'' to that category. To define the addition law, we will use the convention that for a biproduct, ''pk'' will denote the projection morphisms, and ''ik'' will denote the injection morphisms. The ''diagonal morphism'' is the canonical morphism , induced by the universal property of products, such that for . Dually, the ''codiagonal morphism'' is the canonical morphism , induced by the universal property of coproducts, such that for . For each object , we define: * the addition of the injections to be the diagonal morphism, that is ; * the addition of the projections to be the codiagonal morphism, that is . Next, given two morphisms , there exists a unique morphism such that equals if , and 0 otherwise. We can therefore define . This addition is both commutative and associative. The associativity can be seen by considering the composition :A\ \xrightarrow\ A \oplus A \oplus A\ \xrightarrow\ B \oplus B \oplus B\ \xrightarrow\ B We have , using that . It is also bilinear, using for example that and that . We remark that for a biproduct we have . Using this, we can represent any morphism as a matrix.


Matrix representation of morphisms

Given objects and in an additive category, we can represent morphisms as -by- matrices :\begin f_ & f_ & \cdots & f_ \\ f_ & f_ & \cdots & f_ \\ \vdots & \vdots & \cdots & \vdots \\ f_ & f_ & \cdots & f_ \end where f_ := p_k \circ f \circ i_l\colon A_l \to B_k. Using that , it follows that addition and composition of matrices obey the usual rules for matrix addition and
multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
. Thus additive categories can be seen as the most general context in which the algebra of matrices makes sense. Recall that the morphisms from a single object  to itself form the endomorphism ring . If we denote the -fold product of  with itself by , then morphisms from to are ''m''-by-''n'' matrices with entries from the ring . Conversely, given any ring , we can form a category  by taking objects ''An'' indexed by the set of
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s (including 0) and letting the hom-set of morphisms from to be the set of -by- matrices over , and where composition is given by matrix multiplication. Then is an additive category, and equals the -fold power . This construction should be compared with the result that a ring is a preadditive category with just one object, shown here. If we interpret the object as the left module , then this ''matrix category'' becomes a subcategory of the category of left modules over . This may be confusing in the special case where or is zero, because we usually don't think of matrices with 0 rows or 0 columns. This concept makes sense, however: such matrices have no entries and so are completely determined by their size. While these matrices are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object. Thinking about such matrices can be useful in one way, though: they highlight the fact that given any objects and in an additive category, there is exactly one morphism from to 0 (just as there is exactly one 0-by-1 matrix with entries in ) and exactly one morphism from 0 to (just as there is exactly one 1-by-0 matrix with entries in ) – this is just what it means to say that 0 is a zero object. Furthermore, the zero morphism from to is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.


Additive functors

A
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
between preadditive categories is ''additive'' if it is an abelian group
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
on each hom-set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams. That is, if is a biproduct of  in C with projection morphisms and injection morphisms , then should be a biproduct of  in D with projection morphisms and injection morphisms . Almost all functors studied between additive categories are additive. In fact, it is a theorem that all adjoint functors between additive categories must be additive functors (see here). Most of the interesting functors studied in category theory are adjoints.


Generalization

When considering functors between -linear additive categories, one usually restricts to - linear functors, so those functors giving an - module homomorphism on each hom-set.


Special cases

* A '' pre-abelian category'' is an additive category in which every morphism has a kernel and a cokernel. * An ''
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 prototypical example of an abelian category is the category o ...
'' is a pre-abelian category such that every monomorphism and epimorphism is normal. Many commonly studied additive categories are in fact abelian categories; for example, Ab is an abelian category. The free abelian groups provide an example of a category that is additive but not abelian..


References

* Nicolae Popescu; 1973; ''Abelian Categories with Applications to Rings and Modules''; Academic Press, Inc. (out of print) goes over all of this very slowly {{Category theory