Editing Additive category

{{Short description|Preadditive category that admits all finitary products}}
{{for|the weaker sense of the term "additive category" (without biproducts)|preadditive category}}
In [[mathematics]], specifically in [[category theory]], an '''additive category''' is a [[preadditive category]]&nbsp;'''C''' admitting all [[finitary]] [[biproduct]]s.

== Definition ==
There are two equivalent definitions of an additive category: One as a [[category (mathematics)|category]] equipped with additional structure, and another as a category equipped with ''no extra structure'' but whose objects and [[morphism]]s satisfy certain equations.

=== Via preadditive categories ===
A category '''C''' is preadditive if all its [[hom-set]]s are [[abelian group]]s and composition of morphisms is [[bilinear map|bilinear]]; in other words, '''C''' is [[enriched category|enriched]] over the [[monoidal category]] of abelian groups.

In a preadditive category, every finitary [[product (category theory)|product]] is necessarily a [[coproduct]], and hence a [[biproduct]], and [[converse (logic)|converse]]ly every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). The empty product, is  a [[Initial_and_terminal_objects|final object]] and  the empty product in the case of an empty diagram, an [[Initial_and_terminal_objects|initial object]]. 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 [[biproduct]]s. It is then a remarkable theorem that the Hom sets naturally admit an [[abelian monoid]] structure. A [[mathematical proof|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]]. This then gives the Hom sets an [[abelian group]] structure instead of merely an abelian monoid structure.

=== Generalization ===

More generally, one also considers additive [[Preadditive category#R-linear categories|{{mvar|R}}-linear categories]] for a [[commutative ring]] {{mvar|R}}. These are categories enriched over the monoidal category of {{mvar|R}}-[[module (mathematics)|module]]s and admitting all finitary biproducts.

== Examples ==

The original example of an additive category is the [[category of abelian groups]] '''Ab'''. The zero object is the [[trivial group]], the addition of morphisms is given [[pointwise]], and biproducts are given by [[direct sum of abelian groups|direct sums]].

More generally, every [[module category]] over a [[ring (mathematics)|ring]] {{mvar|R}} is additive, and so in particular, the [[category of vector spaces]] over a [[field (mathematics)|field]] {{mvar|K}} is additive.

The algebra of [[matrix (mathematics)|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#Commutative monoid|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]] every morphism has an additive inverse.

This shows that the addition law for an additive category is ''internal'' to that category.<ref>{{citation|mr=0049192|first=Saunders|last=MacLane|title=Duality for groups|journal=Bulletin of the American Mathematical Society|volume=56|issue=6|year=1950|pages=485–516
|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183515045|doi=10.1090/S0002-9904-1950-09427-0|doi-access=free}} Sections 18 and 19 deal with the addition law in semiadditive categories.</ref>

To define the addition law, we will use the convention that for a biproduct, ''p<sub>k</sub>'' will denote the projection morphisms, and ''i<sub>k</sub>'' will denote the injection morphisms.

The ''diagonal morphism'' is the canonical morphism {{math|∆: ''A'' → ''A'' ⊕ ''A''}}, induced by the universal property of products, such that {{math|1=''p''<sub>''k''</sub> ∘ ∆ = 1<sub>''A''</sub>}} for {{math|1=''k'' = 1, 2}}. Dually, the ''codiagonal morphism'' is the canonical morphism {{math|∇: ''A'' ⊕ ''A'' → ''A''}}, induced by the universal property of coproducts, such that {{math|1=∇ ∘ ''i''<sub>''k''</sub> = 1<sub>''A''</sub>}} for {{math|1=''k'' = 1, 2}}.

For each object {{mvar|A}}, we define:

* the addition of the injections {{math|''i''<sub>1</sub> + ''i''<sub>2</sub>}} to be the diagonal morphism, that is {{math|1=∆ = ''i''<sub>1</sub> + ''i''<sub>2</sub>}};
* the addition of the projections {{math|''p''<sub>1</sub> + ''p''<sub>2</sub>}} to be the codiagonal morphism, that is {{math|1=∇ = ''p''<sub>1</sub> + ''p''<sub>2</sub>}}.

Next, given two morphisms {{math|α<sub>''k''</sub>: ''A'' → ''B''}}, there exists a unique morphism {{math|α<sub>1</sub> ⊕ α<sub>2</sub>: ''A'' ⊕ ''A'' → ''B'' ⊕ ''B''}} such that {{math|''p''<sub>''l''</sub> ∘ (α<sub>1</sub> ⊕ α<sub>2</sub>) ∘ ''i''<sub>''k''</sub>}} equals {{math|α<sub>''k''</sub>}} if {{math|1=''k'' = ''l''}}, and 0 otherwise.

We can therefore define {{math|1=α<sub>1</sub> + α<sub>2</sub> := ∇ ∘ (α<sub>1</sub> ⊕ α<sub>2</sub>) ∘ ∆}}.

This addition is both commutative and associative. The associativity can be seen by considering the composition
:<math>A\ \xrightarrow{\quad\Delta\quad}\ A \oplus A \oplus A\ \xrightarrow{\alpha_1\,\oplus\,\alpha_2\,\oplus\,\alpha_3}\ B \oplus B \oplus B\ \xrightarrow{\quad\nabla\quad}\ B</math>

We have {{math|1=α + 0 = α}}, using that {{math|1=α ⊕ 0 = ''i''<sub>1</sub> ∘ α ∘ ''p''<sub>1</sub>}}.

It is also bilinear, using for example that {{math|1=∆ ∘ β = (β ⊕ β) ∘ ∆}} and that {{math|1=(α<sub>1</sub> ⊕ α<sub>2</sub>) ∘ (β<sub>1</sub> ⊕ β<sub>2</sub>) = (α<sub>1</sub> ∘ β<sub>1</sub>) ⊕ (α<sub>2</sub> ∘ β<sub>2</sub>)}}.

We remark that for a biproduct {{math|''A'' ⊕ ''B''}} we have {{math|1=''i''<sub>1</sub> ∘ ''p''<sub>1</sub> + ''i''<sub>2</sub> ∘ ''p''<sub>2</sub> = 1}}. Using this, we can represent any morphism {{math|''A'' ⊕ ''B'' → ''C'' ⊕ ''D''}} as a matrix.

== Matrix representation of morphisms ==

Given objects {{math|''A''<sub>1</sub>, ..., ''A<sub>n</sub>''}} and {{math|''B''<sub>1</sub>, ..., ''B<sub>m</sub>''}} in an additive category, we can represent morphisms {{math|''f'': ''A''<sub>1</sub> ⊕ ⋅⋅⋅ ⊕ ''A<sub>n</sub>'' → ''B''<sub>1</sub> ⊕ ⋅⋅⋅ ⊕ ''B<sub>m</sub>''}} as {{mvar|m}}-by-{{mvar|n}} matrices
:<math>\begin{pmatrix} f_{11} & f_{12} & \cdots & f_{1n} \\
f_{21} & f_{22} & \cdots & f_{2n} \\
\vdots & \vdots & \cdots & \vdots \\
f_{m1} & f_{m2} & \cdots & f_{mn} \end{pmatrix}
</math> where <math>f_{kl} := p_k \circ f \circ i_l\colon A_l \to B_k.</math>

Using that {{math|1=∑<sub>''k''</sub> ''i''<sub>''k''</sub> ∘ ''p''<sub>''k''</sub> = 1}}, it follows that addition and composition of matrices obey the usual rules for [[matrix addition]] and [[matrix multiplication|multiplication]].

<!--
use [[matrix (mathematics)|matrices]] to study the biproducts of ''A'' and ''B'' with themselves.
Specifically, if we define the ''biproduct power'' ''A<sup>n</sup>'' to be the ''n''-fold biproduct ''A'' ⊕ ⋯ ⊕ ''A'' and ''B<sup>m</sup>'' similarly, then the morphisms from ''A<sup>n</sup>'' to ''B<sup>m</sup>'' can be described as ''m''-by-''n'' matrices whose entries are morphisms from ''A'' to ''B''.

For a concrete example, consider the category of [[real number|real]] [[vector space]]s, so that ''A'' and ''B'' are individual vector spaces.
(There is no need for ''A'' and ''B'' to have [[finite set|finite]] [[dimension (mathematics)|]]s, although of course the numbers ''m'' and ''n'' must be finite.)
Then an element of ''A''<sup>''n''</sup> may be represented as an ''n''-by-{{num|1}} [[column vector]] whose entries are elements of ''A'':

<math>\begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix}</math>

and a morphism from ''A''<sup>''n''</sup> to ''B''<sup>''m''</sup> is an ''m''-by-''n'' matrix whose entries are morphisms from ''A'' to ''B'':

<math>\begin{pmatrix} f_{1,1} & f_{1,2} & \cdots & f_{1,n} \\
f_{2,1} & f_{2,2} & \cdots & f_{2,n} \\
\vdots & \vdots & \cdots & \vdots \\
f_{m,1} & f_{m,2} & \cdots & f_{m,n} \end{pmatrix}</math>

Then this morphism matrix acts on the column vector by the usual rules of matrix multiplication to give an element of ''B''<sup>''m''</sup>, represented by an ''m''-by-1 column vector with entries from ''B'':

<math>\begin{pmatrix} f_{1,1} & f_{1,2} & \cdots & f_{1,n} \\
f_{2,1} & f_{2,2} & \cdots & f_{2,n} \\
\vdots & \vdots & \cdots & \vdots \\
f_{m,1} & f_{m,2} & \cdots & f_{m,n} \end{pmatrix}
\begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix}
= \begin{pmatrix} f_{1,1}(a_{1}) + f_{1,2}(a_{2}) + \cdots + f_{1,n}(a_{n}) \\
f_{2,1}(a_{1}) + f_{2,2}(a_{2}) + \cdots + f_{2,n}(a_{n}) \\
\cdots \\
f_{m,1}(a_{1}) + f_{m,2}(a_{2}) + \cdots + f_{m,n}(a_{n}) \end{pmatrix}</math>

Even in the setting of an abstract additive category, where it makes no sense to speak of elements of the objects ''A''<sup>''n''</sup> and ''B''<sup>''m''</sup>, the matrix representation of the morphism is still useful, because [[matrix multiplication]] correctly reproduces composition of morphisms.
-->
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&nbsp;{{mvar|A}} to itself form the [[endomorphism ring]] {{math|End&thinsp;''A''}}.
If we denote the {{mvar|n}}-fold product of&nbsp;{{mvar|A}} with itself by {{math|''A''<sup>''n''</sup>}}, then morphisms from {{math|''A<sup>n</sup>''}} to {{math|''A<sup>m</sup>''}} are ''m''-by-''n'' matrices with entries from the ring {{math|End&thinsp;''A''}}.

Conversely, given any [[ring (mathematics)|ring]] {{mvar|R}}, we can form a category&nbsp;{{math|'''Mat'''(''R'')}} by taking objects ''A<sub>n</sub>'' indexed by the set of [[natural number]]s (including [[0]]) and letting the hom-set of morphisms from {{math|''A<sub>n</sub>''}} to {{math|''A<sub>m</sub>''}} be the [[set (mathematics)|set]] of {{mvar|m}}-by-{{mvar|n}} matrices over&nbsp;{{mvar|R}}, and where composition is given by matrix multiplication.<ref>H.D. Macedo, J.N. Oliveira, [https://hal.inria.fr/hal-00919866 Typing linear algebra: A biproduct-oriented approach], Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191, {{issn|0167-6423}}, {{doi|10.1016/j.scico.2012.07.012}}.</ref> Then {{math|'''Mat'''(''R'')}} is an additive category, and {{math|''A''<sub>''n''</sub>}} equals the {{mvar|n}}-fold power {{math|(''A''<sub>1</sub>)<sup>''n''</sup>}}.

This construction should be compared with the result that a ring is a preadditive category with just one object, shown [[Preadditive category#Special cases|here]].

If we interpret the object {{math|''A''<sub>''n''</sub>}} as the left [[module (mathematics)|module]]&nbsp;{{math|''R''<sup>''n''</sup>}}, then this ''matrix category'' becomes a [[subcategory]] of the category of left modules over&nbsp;{{mvar|R}}.

This may be confusing in the special case where {{mvar|m}} or {{mvar|n}} is zero, because we usually don't think of [[empty matrix|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 {{mvar|A}} and {{mvar|B}} in an additive category, there is exactly one morphism from {{mvar|A}} to 0 (just as there is exactly one 0-by-1 matrix with entries in {{math|End&thinsp;''A''}}) and exactly one morphism from 0 to {{mvar|B}} (just as there is exactly one 1-by-0 matrix with entries in {{math|End&thinsp;''B''}})&nbsp;– this is just what it means to say that 0 is a [[zero object (algebra)|zero object]]. Furthermore, the zero morphism from {{mvar|A}} to {{mvar|B}} is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.

== Additive functors ==

A [[functor]] {{math|''F'': '''C''' → '''D'''}} between preadditive categories is ''additive'' if it is an abelian group [[homomorphism]] 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 {{mvar|B}} is a biproduct of&nbsp;{{math|''A''<sub>1</sub>, ... , ''A<sub>n</sub>''}} in&nbsp;'''C''' with projection morphisms {{math|''p<sub>k</sub>''}} and injection morphisms {{math|''i<sub>j</sub>''}}, then {{math|''F''(''B'')}} should be a biproduct of&nbsp;{{math|''F''(''A''<sub>1</sub>), ... , ''F''(''A<sub>n</sub>'')}} in&nbsp;'''D''' with projection morphisms {{math|''F''(''p''<sub>''j''</sub>)}} and injection morphisms {{math|''F''(''i<sub>j</sub>'')}}.

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 [[Adjoint functors#Additivity|here]]). Most of the interesting functors studied in category theory are adjoints.

=== Generalization ===

When considering functors between {{mvar|R}}-linear additive categories, one usually restricts to {{mvar|R}}-[[preadditive category#R-linear categories|linear functors]], so those functors giving an {{mvar|R}}-[[module homomorphism]] on each hom-set.

== Special cases ==

* A ''[[pre-abelian category]]'' is an additive category in which every morphism has a [[kernel (category theory)|kernel]] and a [[cokernel (category theory)|cokernel]].
* An ''[[abelian category]]'' is a pre-abelian category such that every [[monomorphism]] and [[epimorphism]] is [[normal monomorphism|normal]].
Many commonly studied additive categories are in fact abelian categories; for example, '''Ab''' is an abelian category. The [[free abelian group]]s provide an example of a category that is additive but not abelian.<ref>{{citation|title=Basic Algebraic Topology|first=Anant R.|last=Shastri|publisher=CRC Press|year=2013|isbn=9781466562431|page=466|url=https://books.google.com/books?id=lYMAAQAAQBAJ&pg=PA466}}.</ref>

== References ==

{{reflist|1}}

* [[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}}

[[Category:Additive categories| ]]