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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nA *skeleton* of $\\mathcal{C}$ is a [[full subcategory|V3BFGKBJF7]] $\\mathcal{S}$ of $\\mathcal{C}$ such that every object of $\\mathcal{C}$ is [[isomorphic|859F0TFVA3]] to exactly one object of $\\mathcal{S}$.\n\n## Remarks\n\nA category is *skeletal* if isomorphic objects in it are equal; a skeleton is a skeletal full subcategory that meets every isomorphism class.\nUnder the axiom of choice every category has a skeleton, and its inclusion is fully faithful and essentially surjective.\nSo the inclusion is an [[equivalence of categories|K7A04ZBD92]] by [[the criterion for equivalences|WA60RW9E04]].\nAny two skeletons of $\\mathcal{C}$ are isomorphic categories.\nExample: in the category of finite sets, the full subcategory on the sets $[n] = \\set{1, \\dots, n}$, $n \\ge 0$, with $[0] = \\emptyset$, is a skeleton.\nA finite set with $n$ elements is in bijection with $[n]$, and $[m]$ and $[n]$ are not in bijection when $m \\ne n$.\nNon-example: the full subcategory on the sets $[n]$ together with $\\set{a, b}$ is not a skeleton, since $\\set{a, b}$ and $[2]$ are two of its objects in the same isomorphism class.\n",
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