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  "body": "\n## Definition\n\nLet $\\lambda$ be a [[regular cardinal|13ZV2F8JAR]].\nA small category $\\mathcal{I}$ is *$\\lambda$-filtered* if every subcategory of $\\mathcal{I}$ with fewer than $\\lambda$ morphisms has a cocone in $\\mathcal{I}$.\nLet $\\mathcal{C}$ be a locally small [[category|A2DE66937D]].\nAn object $A$ of $\\mathcal{C}$ is *$\\lambda$-presentable* if the [[hom functor|VJXS8AK67E]] $\\mathcal{C}(A, -)$ preserves [[colimits|BJS1CGTKQ5]] of $\\lambda$-filtered diagrams.\nThe category $\\mathcal{C}$ is *$\\lambda$-accessible* if both conditions hold.\n\n1. Every diagram $\\mathcal{I} \\to \\mathcal{C}$ with $\\mathcal{I}$ small and $\\lambda$-filtered has a colimit in $\\mathcal{C}$.\n2. There is a set $S$ of $\\lambda$-presentable objects such that every object of $\\mathcal{C}$ is a colimit of a $\\lambda$-filtered diagram of objects of $S$.\n\nThe category $\\mathcal{C}$ is *accessible* if it is $\\lambda$-accessible for some regular cardinal $\\lambda$.\n\n## Remarks\n\nFor $\\lambda = \\aleph_0$, the $\\lambda$-filtered categories are the [[filtered categories|975PMY4FG8]], and one says finitely presentable and finitely accessible.\n\nExample: the category of sets is finitely accessible.\nIt has all colimits, so clause 1 holds.\nA finite set $A$ is finitely presentable, since a map from $A$ into a filtered colimit factors through some stage, essentially uniquely.\nEvery set is the directed union, hence the filtered colimit, of its finite subsets, and the finite sets $\\set{1, \\dots, n}$ form a set $S$ up to isomorphism.\n\nNon-example: let $\\mathcal{M}$ be the category with one object $*$ and morphisms $1$ and $e$, where $e \\circ e = e$.\nThe identity diagram of $\\mathcal{M}$ is $\\lambda$-filtered for every $\\lambda$, with the cocone $e$.\nIt has no colimit: the cocone $e$ is the only one, and both $1$ and $e$ factor it through itself.\nSo clause 1 fails for every $\\lambda$, and $\\mathcal{M}$ is not accessible.\n",
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