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  "body": "\n## Example\n\nThe *category of abelian groups* $\\mathbf{Ab}$ has the [[abelian groups|082GHNPJ53]] as objects and the [[group homomorphisms|0JYTG5K9Y4]] between them as morphisms, with composition of maps.\nIt is an [[abelian category|Y9CZ3MTNF1]].\nEach set $\\Hom(A, B)$ is an abelian group under pointwise addition $(f + g)(a) = f(a) + g(a)$, and composition is bilinear.\nThe trivial group is a [[zero object|AQET9YHH6F]], and $A \\oplus B$ is both the product and the coproduct of $A$ and $B$.\nEvery homomorphism $f \\colon A \\to B$ has a [[kernel|S417YNEJ6M]] $\\set{a \\in A : f(a) = 0}$ and a [[cokernel|H8W38933NC]] $B / f(A)$.\nAn injective homomorphism $f$ is the kernel of the projection $B \\to B/f(A)$.\nA surjective homomorphism is the cokernel of the inclusion of its kernel.\nSo every monomorphism is a kernel and every epimorphism is a cokernel.\n\n## Remarks\n\n$\\mathbf{Ab}$ is the model case of an abelian category: the [[Freyd–Mitchell embedding theorem|8V296XR2M3]] says that each small abelian category embeds fully and exactly into a category of modules.\nIt is also a full subcategory of the [[category of groups|2CX7RPPXA9]], but the coproducts differ: in $\\mathbf{Ab}$ the coproduct of $\\Z$ and $\\Z$ is $\\Z \\oplus \\Z$, not the free product.\nThe group $\\Z$ is a [[generator|CK08JW32WB]], and $\\Q/\\Z$ is an [[injective object|5QRFTBF4PG]].\n$\\mathbf{Ab}$ is the same as the [[category of modules|JBSSXP4FFY]] over the ring $\\Z$.\n\n## Explanations\n\n### L1 human\n\nIn the category of abelian groups, maps between two groups can themselves be added, because the groups are commutative.\nThis extra structure lets one talk about kernels, images and exact sequences, which is the starting point of homological algebra.\nFor example, multiplication by $2$ on $\\Z$ is a morphism with kernel $0$ and cokernel $\\Z/2\\Z$.\n",
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