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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 groupsIn mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. The category Grp is both complete and co-complete.. 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:
The third equality requires the group to be abelian. This addition of morphism turns Ab into a preadditive categoryA preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinea, and because the direct sumIn abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. In a sense, the direct sum of vector spaces is the "most general" vector space that contains the of finitely many abelian groups yields a biproductIn category theory and its applications to mathematics, a biproduct is a generalisation of the notion of direct sum that makes sense in any preadditive category. Definition Let C be a preadditive category. In particular, morphisms in C can be added. Given, we indeed have an additive categoryCategory theory In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A,. A of C have a biproduct A ⊕ ··· ⊕ A in C. Recall that a category C is preadditive if all.
In Ab, the notion of kernel in the category theory senseIn category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the mor coincides with kernel in the algebraic senseAlgebra In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. The definition of kernel takes various forms in various conte, 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 of groups; 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 Ab5 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).
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 strict monoidal category.
Ab is not cartesian closed (and therefore also not a topos) since it lacks exponential object s.
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