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In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab.That is, an Ab-category C is a category such thatevery hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation.In formulas: andwhere + is the group operation.

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rdf:type	yago:Abstraction100002137 yago:Class107997703 yago:Collection107951464 yago:Group100031264 television station yago:WikicatAdditiveCategories
rdfs:label	前加法圏 (ja) Pre-additieve categorie (nl) Preadditive category (en) Предаддитивная категория (ru) 預可加範疇 (zh)
rdfs:comment	数学、特に圏論において、前加法圏とは可換群のなすモノイド圏で豊穣化した圏のことである。言い換えると、圏Cが前加法的であるとは、Cの各 Hom(A,B) が可換群の構造を持ち、さらに射の合成について双線形であることをいう。可換群の圏を Ab と書く記法に由来して、前加法圏を「Ab-圏」と呼ぶこともある。著者によっては前加法圏を加法圏と呼ぶこともあるが、ある特別な前加法圏(以下のを参照)のことを加法圏と呼ぶのが最近の傾向である。 (ja) In de categorietheorie, een abstract deelgebied van de wiskunde, is een pre-additieve categorie een categorie, die is verrijkt over de monoïdale categorie van abelse groepen. Met andere woorden de categorie C is pre-additief als iedere Hom(A,B) in C de structuur van een abelse groep heeft, en de samenstelling van morfismen is over de gehele getallen. Een pre-additieve categorie wordt ook wel een Ab-categorie genoemd. De notatie Ab staat voor de categorie van abelse groepen. Sommige auteurs hebben de term additieve categorie gebruikt voor pre-additieve categorieën. (nl) Предаддити́вная категория — обогащённая категория над категорией абелевых групп, то есть такая категория, что для любых её объектов , множество имеет структуру абелевой группы по сложению, при этом композиция морфизмов билинейна: Предаддитивную категорию иногда называют также -категорией. (ru) 在範疇論中，一個預可加範疇是使得任兩個對象間的態射集帶有交換群結構，並使得態射合成為雙線性運算之範疇。形式地說，預可加範疇是在交換群的么半範疇上的範疇。預加法範疇有時亦稱Ab-範疇，其中的Ab是交換群範疇的縮寫。舊文獻有時也將預加法範疇稱為加法範疇；在此則採當代觀點，區別預加法範疇與可加範疇。一般而言，固定一個交換環，我們可以定義-預可加範疇為在-模的么半範疇上濃化的範疇，即：使任兩個對象間的態射集為-模，並使態射合成為上的雙線性運算之範疇。取則回到原始定義。 (zh) In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab.That is, an Ab-category C is a category such thatevery hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation.In formulas: andwhere + is the group operation. (en)
dcterms:subject	Additive categories
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Wikipage revision ID	1095242365 (xsd:integer)
Link from a Wikipage to another Wikipage	Endomorphism ring Enriched category Epimorphism Module (mathematics) Monomorphism Normal monomorphism Bilinear operator Hom-set Jacobson radical Pre-abelian category Trivial group 0 Commutativity Mathematics Matrix (mathematics) Natural transformation Coequaliser Endomorphism Function (mathematics) Monoid Coproduct Equaliser (mathematics) Closed monoidal category Commutative ring Zero object Functor Functor category Identity element Kernel (algebra) 1 Additive category Field (mathematics) Finite set Nicolae Popescu Charles Weibel Direct sum Product (category theory) Ring (mathematics) Group (mathematics) Group homomorphism Terminal object Zero morphism Ring (algebra) Abelian category Additive categories Abstract nonsense Kernel (category theory) Cokernel Distinct (mathematics) Distributivity Enriched functor Factor ring Identity morphism If and only if Infinity Initial object Category (mathematics) Category of abelian groups Category of modules Category theory Category of vector spaces Ring homomorphism Ideal (ring) Medial category Trivial ring
sameAs	Preadditive category Preadditive category Preadditive category Preadditive category Preadditive category Preadditive category Preadditive category Preadditive category
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has abstract	In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab.That is, an Ab-category C is a category such thatevery hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation.In formulas: andwhere + is the group operation. Some authors have used the term additive category for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below). (en)

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