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...says that a functor <math>F : \mathbf{Mod}_R \to \mathbf{Mod}_S</math> is additive, is right-exact and preserves coproducts if and only if it is of the form < *Charles E. Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11, 1960, 5–8. ...

==Left and right semi-abelian categories== ...f</math>. Accordingly, '''right semi-abelian categories''' are pre-abelian categories such that <math>\overline{f}</math> is an epimorphism for each morphism <ma ...

...tor''' is an [[additive functor]] ''F'' between [[abelian category|abelian categories]] ''C'' and ''D'' for which, for each object ''A'' in ''C'', there exists a ...

...aunders Mac Lane | last = Laplaza| first = Miguel L.| title = Coherence in Categories| chapter = Coherence for distributivity| series = Lecture Notes in Mathemat ...structure is the coproduct are called [[distributive category|distributive categories]]. ...

...anch of mathematics, a '''Krull–Schmidt category''' is a generalization of categories in which the [[Krull–Schmidt theorem]] holds. They arise, for example, in t ...category]], or more generally an additive [[preadditive category#R-linear categories|{{mvar|R}}-linear category]] for a [[commutative ring]]&nbsp;{{mvar|R}}. We ...

* ''A''(''G'') is an [[additive category]] with direct sums given by the disjoint union of ''G''-sets and z ...tac.mta.ca/tac/volumes/38/6/38-06.pdf |journal=[[Theory and Application of Categories]] |volume=38 |issue=6 |pages=158}}</ref> ...

...r]] to the product are called [[Cartesian closed category|Cartesian closed categories]].<ref>{{nlab|id=cartesian+monoidal+category|title=Cartesian monoidal categ Cartesian monoidal categories have a number of special and important properties, such as the existence of ...

...name="exact">{{cite journal |last1=Positselski |first1=Leonid |title=Exact categories of topological vector spaces with linear topology |journal=Moscow Math. Jou ==Left and right quasi-abelian categories== ...

== Connection with the basic decomposition in pre-abelian categories == == Categories with nodal decomposition == ...

...sentable by a finite flat ''S''-group scheme, and Cartier duality forms an additive involutive antiequivalence from the category of finite flat commutative ''S ...iequivalence with commutative affine [[formal group]]s, so if ''G'' is the additive group <math>\mathbf{G'}_a</math>, then its Cartier dual is the multiplicati ...

...nd |first=Tom |date=2006-02-08 |title=Stability conditions on triangulated categories |eprint=math/0212237 }}</ref> ...esented as in the original paper of Bridgeland, for arbitrary triangulated categories.<ref name=bridgeland /> Let <math>\mathcal{D}</math> be a triangulated cate ...

...ace''' is a [[Banach lattice]] <math>(X, \| \cdot \|)</math> whose norm is additive on the positive cone of ''X''.{{sfn | Schaefer | Wolff | 1999 | pp=242–250} <!--- Categories ---> ...

...k'' of characteristic zero): any morphism (of algebraic groups) from the [[additive group]] <math>G_a</math> to a [[reductive group]] ''H'' factors through the ...joint]], the so-called pro-reductive envelope. This left adjoint sends the additive group <math>G_a</math> to <math>SL_2</math> (which happens to be semi-simpl ...

In [[homological algebra]], a '''δ-functor''' between two [[abelian categories]] ''A'' and ''B'' is a collection of [[functor]]s from ''A'' to ''B'' toget ...'' and ''B''''' is a family {''T''ⁿ} of [[functor|covariant]] [[additive functor]]s ''T''ⁿ : ''A'' → ''B'' [[Indexed family|indexed]] by ...

...r [[sheaf (mathematics)|sheaves]] in a unified manner. The theory of these categories was further developed in [[Pierre Gabriel]]'s 1962 thesis.<ref name="gabrie ...utative varieties" is then nothing but the study of (certain) Grothendieck categories.<ref>{{citation|author1-first=Hoang Dinh|author1-last=Van|author2-first=Liy ...

...[[lek paradox]] by proposing that recurrent deleterious mutation maintains additive genetic variance in fitness by incorporating the entire [[mutation load]] o <!--- Categories ---> ...

...h>-category (<math>(\infty, 1)</math>-categories are also known as [[quasi-categories]]). A Mackey functor is a [[product-preserving functor]] <math>M: \text{Spa ...

==== Additive group ==== Let <math>G = \mathbb{G}_a</math> be the additive group; i.e., ''G''(''R'') = ''R'' for any ''k''-algebra ''R''. As a variety ...

...t that every morphism have a kernel and cokernel, as is true for [[abelian categories]]. is in fact an additive morphism. ...

...raz|date=2011-06-28|title=Cohomology with coefficients in stacks of Picard categories|class=math.AT|eprint=1101.2918}}</ref> which were originally introduced by ...a line bundle are given by a non-vanishing function on <math>X</math>. The additive structure <math>+</math> is given by the tensor product <math>\otimes</math ...