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- Category_of_abelian_groups abstract "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.The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.The zero object of Ab is the trivial group {0} which consists only of its neutral element.Note that Ab is a full subcategory of Grp, the category of all groups. The main difference between Ab and Grp is that the sum of two homomorphisms f and g between abelian groups is again a group homomorphism:(f+g)(x+y) = f(x+y) + g(x+y) = f(x) + f(y) + g(x) + g(y) = f(x) + g(x) + f(y) + g(y) = (f+g)(x) + (f+g)(y)The third equality requires the group to be abelian. This addition of morphism turns Ab into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.In Ab, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e.: the kernel of the morphism f : A → B is the subgroup K of A defined by K = {x in A : f(x) = 0}, together with the inclusion homomorphism i : K → A. The same is true for cokernels: the cokernel of f is the quotient group C = B/f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp: in Grp it can happen that f(A) is not a normal subgroup of B, and that therefore the quotient group B/f(A) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an abelian category. The product in Ab is given by the product of groups, formed by taking the cartesian product of the underlying sets and performing the group operation componentwise. Because Ab has kernels, one can then show that Ab is a complete category. The coproduct in Ab is given by the direct sum; since Ab has cokernels, it follows that Ab is also cocomplete. Taking direct limits in Ab is an exact functor, which turns Ab into an abelian category.We have a forgetful functor Ab → Set which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore Ab is a concrete category. The forgetful functor has a left adjoint (which associates to a given set the free abelian group with that set as basis) but does not have a right adjoint.An object in Ab is injective if and only if it is divisible; it is projective if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator (Q/Z). This implies that Ab is an example of a Grothendieck category.Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an abelian group. With this notion of product, Ab is a symmetric monoidal category.Ab is not cartesian closed (and therefore also not a topos) since it lacks exponential objects.".
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- Category_of_abelian_groups wikiPageLength "5307".
- Category_of_abelian_groups wikiPageOutDegree "63".
- Category_of_abelian_groups wikiPageRevisionID "667770057".
- Category_of_abelian_groups wikiPageWikiLink Abelian_category.
- Category_of_abelian_groups wikiPageWikiLink Abelian_group.
- Category_of_abelian_groups wikiPageWikiLink Abelian_sheaf.
- Category_of_abelian_groups wikiPageWikiLink Additive_category.
- Category_of_abelian_groups wikiPageWikiLink Adjoint_functors.
- Category_of_abelian_groups wikiPageWikiLink Bijection.
- Category_of_abelian_groups wikiPageWikiLink Bijective.
- Category_of_abelian_groups wikiPageWikiLink Biproduct.
- Category_of_abelian_groups wikiPageWikiLink Cambridge_University_Press.
- Category_of_abelian_groups wikiPageWikiLink Cartesian_closed_category.
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- Category_of_abelian_groups wikiPageWikiLink Category:Category-theoretic_categories.
- Category_of_abelian_groups wikiPageWikiLink Category:Group_theory.
- Category_of_abelian_groups wikiPageWikiLink Category_(mathematics).
- Category_of_abelian_groups wikiPageWikiLink Category_of_groups.
- Category_of_abelian_groups wikiPageWikiLink Category_of_modules.
- Category_of_abelian_groups wikiPageWikiLink Category_of_sets.
- Category_of_abelian_groups wikiPageWikiLink Category_theory.
- Category_of_abelian_groups wikiPageWikiLink Cocomplete.
- Category_of_abelian_groups wikiPageWikiLink Cokernel.
- Category_of_abelian_groups wikiPageWikiLink Complete_category.
- Category_of_abelian_groups wikiPageWikiLink Concrete_category.
- Category_of_abelian_groups wikiPageWikiLink Coproduct.
- Category_of_abelian_groups wikiPageWikiLink Direct_limit.
- Category_of_abelian_groups wikiPageWikiLink Direct_product_of_groups.
- Category_of_abelian_groups wikiPageWikiLink Direct_sum.
- Category_of_abelian_groups wikiPageWikiLink Direct_sum_of_abelian_groups.
- Category_of_abelian_groups wikiPageWikiLink Divisible_group.
- Category_of_abelian_groups wikiPageWikiLink Epimorphism.
- Category_of_abelian_groups wikiPageWikiLink Exact_functor.
- Category_of_abelian_groups wikiPageWikiLink Exponential_object.
- Category_of_abelian_groups wikiPageWikiLink Faithful_functor.
- Category_of_abelian_groups wikiPageWikiLink Forgetful_functor.
- Category_of_abelian_groups wikiPageWikiLink Free_abelian_group.
- Category_of_abelian_groups wikiPageWikiLink Full_and_faithful_functors.
- Category_of_abelian_groups wikiPageWikiLink Full_subcategory.
- Category_of_abelian_groups wikiPageWikiLink Function_(mathematics).
- Category_of_abelian_groups wikiPageWikiLink Graduate_Texts_in_Mathematics.
- Category_of_abelian_groups wikiPageWikiLink Grothendieck_category.
- Category_of_abelian_groups wikiPageWikiLink Group_homomorphism.
- Category_of_abelian_groups wikiPageWikiLink Identity_element.
- Category_of_abelian_groups wikiPageWikiLink Initial_and_terminal_objects.
- Category_of_abelian_groups wikiPageWikiLink Injective.
- Category_of_abelian_groups wikiPageWikiLink Injective_cogenerator.
- Category_of_abelian_groups wikiPageWikiLink Injective_function.
- Category_of_abelian_groups wikiPageWikiLink Injective_module.
- Category_of_abelian_groups wikiPageWikiLink Isomorphism.
- Category_of_abelian_groups wikiPageWikiLink Kernel_(algebra).
- Category_of_abelian_groups wikiPageWikiLink Kernel_(category_theory).
- Category_of_abelian_groups wikiPageWikiLink Mathematics.
- Category_of_abelian_groups wikiPageWikiLink Monoidal_category.
- Category_of_abelian_groups wikiPageWikiLink Monomorphism.
- Category_of_abelian_groups wikiPageWikiLink Morphism.
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- Category_of_abelian_groups wikiPageWikiLink Normal_subgroup.
- Category_of_abelian_groups wikiPageWikiLink Object_(category_theory).
- Category_of_abelian_groups wikiPageWikiLink Preadditive_category.
- Category_of_abelian_groups wikiPageWikiLink Product_(category_theory).
- Category_of_abelian_groups wikiPageWikiLink Projective_module.
- Category_of_abelian_groups wikiPageWikiLink Quotient_group.
- Category_of_abelian_groups wikiPageWikiLink Set_(mathematics).
- Category_of_abelian_groups wikiPageWikiLink Sheaf_of_modules.
- Category_of_abelian_groups wikiPageWikiLink Small_category.
- Category_of_abelian_groups wikiPageWikiLink Springer-Verlag.
- Category_of_abelian_groups wikiPageWikiLink Springer_Science+Business_Media.
- Category_of_abelian_groups wikiPageWikiLink Subcategory.
- Category_of_abelian_groups wikiPageWikiLink Surjective.
- Category_of_abelian_groups wikiPageWikiLink Surjective_function.
- Category_of_abelian_groups wikiPageWikiLink Tensor_product.
- Category_of_abelian_groups wikiPageWikiLink Topos.
- Category_of_abelian_groups wikiPageWikiLink Zero_object.
- Category_of_abelian_groups wikiPageWikiLinkText "Ab".
- Category_of_abelian_groups wikiPageWikiLinkText "Category of abelian groups".
- Category_of_abelian_groups wikiPageWikiLinkText "abelian groups".
- Category_of_abelian_groups wikiPageWikiLinkText "category '''Ab''' of abelian groups and group homomorphisms".
- Category_of_abelian_groups wikiPageWikiLinkText "category of abelian groups".
- Category_of_abelian_groups wikiPageWikiLinkText "category".
- Category_of_abelian_groups hasPhotoCollection Category_of_abelian_groups.
- Category_of_abelian_groups wikiPageUsesTemplate Template:Cite_book.
- Category_of_abelian_groups wikiPageUsesTemplate Template:Lang_Algebra.
- Category_of_abelian_groups wikiPageUsesTemplate Template:Reflist.
- Category_of_abelian_groups subject Category:Category-theoretic_categories.
- Category_of_abelian_groups subject Category:Group_theory.
- Category_of_abelian_groups comment "In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms.".
- Category_of_abelian_groups label "Category of abelian groups".
- Category_of_abelian_groups sameAs Categoría_de_grupos_abelianos.
- Category_of_abelian_groups sameAs Catégorie_des_groupes_abéliens.
- Category_of_abelian_groups sameAs Categorie_van_abelse_groepen.
- Category_of_abelian_groups sameAs Categoria_de_grupos_abelianos.
- Category_of_abelian_groups sameAs Категория_абелевых_групп.
- Category_of_abelian_groups sameAs Q1956793.
- Category_of_abelian_groups sameAs Q1956793.
- Category_of_abelian_groups wasDerivedFrom Category_of_abelian_groups?oldid=667770057.
- Category_of_abelian_groups isPrimaryTopicOf Category_of_abelian_groups.