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  "body": "\n## Definition\n\nLet $\\lambda$ be a [[regular cardinal|13ZV2F8JAR]] and let $\\mathcal{C}$ be a locally small category.\nThe category $\\mathcal{C}$ is *locally $\\lambda$-presentable* if both conditions hold.\n\n1. $\\mathcal{C}$ is $\\lambda$-[[accessible|FC8C0V4TFZ]].\n2. $\\mathcal{C}$ has all small [[colimits|BJS1CGTKQ5]].\n\nThe category $\\mathcal{C}$ is *locally presentable* if it is locally $\\lambda$-presentable for some regular cardinal $\\lambda$.\n\n## Remarks\n\nEquivalently, $\\mathcal{C}$ is cocomplete and has a set $S$ of $\\lambda$-presentable objects such that every object is a $\\lambda$-filtered colimit of objects of $S$.\nFor $\\lambda = \\aleph_0$ one says locally finitely presentable.\n\nExample: the category of sets is locally finitely presentable.\nIt is cocomplete, and it is finitely accessible because every set is the filtered colimit of its finite subsets, which are finitely presentable.\n\nNon-example: the category of fields and field homomorphisms is accessible but not locally presentable.\nIt has no initial object, which is the colimit of the empty diagram.\nAn initial field would map to both $\\Q$ and $\\F_2$, but a field homomorphism preserves the characteristic.\nSo clause 2 fails.\n\nA locally presentable category is also complete, well-powered and co-well-powered.\n",
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