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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nThe category $\\mathcal{C}$ is *well-powered* if for every object $X$ of $\\mathcal{C}$ the [[subobjects|FS1FKAE89H]] of $X$ form a [[set|0BSJ7RQ9BW]], rather than a proper class.\nDually, $\\mathcal{C}$ is *co-well-powered* if for every object $X$ the quotient objects of $X$, the equivalence classes of epimorphisms out of $X$, form a set.\n\n## Remarks\n\nExample: the category of sets is well-powered.\nEach subobject of a set $X$ is represented by the inclusion of a subset.\nSo the subobjects of $X$ are in bijection with the [[power set|00E0YZCERH]] of $X$, which is a set.\nThe same holds for groups, rings and modules, where subobjects correspond to substructures.\nIn the category of topological spaces, a subobject of $X$ corresponds to a subset of $X$ with a topology finer than the subspace topology.\nThese pairs also form a set.\n\nNon-example: let $\\mathcal{C}$ have as objects all sets and one more object $X$.\nBesides identities, let $\\mathcal{C}$ have exactly one morphism $m_S \\colon S \\to X$ for each set $S$, and nothing else.\nIt is a category, since no two non-identity morphisms compose.\nEach $m_S$ is a monomorphism, since there are no two different morphisms into $S$ from one object, and $m_S \\equiv m_T$ only when $S = T$, since there is no morphism between distinct sets.\nSo the subobjects of $X$ form a proper class, and $\\mathcal{C}$ is not well-powered.\n",
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