Pre-Abelian category
Pre-Abelian category In mathematics, specifically in category theory, a pre-Abelian category is an additive category that has ... this means that a category C is pre-Abelian if: C is preadditive, that ...
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User:Lethe/list of categories
User:Lethe/list of categories < User:Lethe In category theory, categories are the main object of study. The following is a list of important categories, and a glossary of named categories. Table of categories Legend symbol meaning keys and comments ...
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List of mathematics categories
List of mathematics categories Mathematics categories The contents of these categories will be automatically added to the list ... manifolds -- 4-dimensional geometry -- 4-manifolds -- A Abelian group theory -- Abelian varieties -- Abstract algebra -- Adjoint functors -- Affine ...
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Preadditive category
... is enriched over the monoidal category of abelian groups. In other words, the category C ... in C has the structure of an abelian group, and composition of morphisms is bilinear ... 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). Examples The ...
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Additive category
... preadditive if all its morphism sets are Abelian groups and morphism composition is bilinear, i ... is enriched over the monoidal category of Abelian groups; and recall that a biproduct in ... additive category is the category Ab of Abelian groups with group homomorphisms. Ab is preadditive ... category, and many basic properties of these categories are described under that subject. This article ... correctly reproduces composition of morphisms. Thus additive categories can be seen as the most ...
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Talk:Additive category
... characterisation. There's an elementary characterisation of Abelian categories, which is pretty slick, and which I ... and there's an elementary characterisation of pre-Abelian categories too. So I wanted to describe ...
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Limit (category theory)
... more aptly called sources. Therefore consider two categories J and C and a covariant functor ... L. In many algebraic contexts, such as (abelian) groups, rings, boolean algebras, etc., products are ... sets that contain the empty set). Complete categories A category C is called complete iff ... said to have finite limits. Many important categories are complete: groups, abelian groups, sets, modules over some ring, ...
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Ring theory
... in the mid-nineteenth century undercut the pre-eminence of fields in mathematical analysis. Richard ... introduction Definition Formally, a ring is an abelian group (R, +), together with a second binary ... vector spaces) are often studied via their categories of modules. A module over a ring is an abelian group that the ring acts on as ... more generally by rings of endomorphisms of abelian groups or modules, and by monoid ...
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List of category theory topics
... category theory topics, by Wikipedia page. Specific categories Category of sets Concrete category Category of ... preordered sets Category of groups Category of abelian groups Category of commutative rings Category of ... Cartesian morphism Slice category Functors Isomorphism of categories Natural transformation Equivalence of categories Subcategory Faithful functor Full functor Forgetful functor ... sum Additive structure Preadditive category Additive category Pre-Abelian category Abelian category Exact sequence ...
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Cokernel
... image of f. In the case of abelian groups, since every subgroup is normal, the ... f, g) = coker(g - f) In a pre-abelian category (a special kind of preadditive category ... kernels and cokernels is guaranteed. In such categories the image and coimage of a morphism ... coker f) coim(f) = coker(ker f) Abelian categories are even better behaved with ...
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