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  "body": "\n## Definition\n\nLet $J$ be a [[category|A2DE66937D]].\n$J$ is *filtered* if:\n\n1. $J$ has at least one object.\n2. For any two objects $a$ and $b$ of $J$, there are an object $c$ and [[morphisms|859F0TFVA3]] $a \\to c$ and $b \\to c$.\n3. For any two morphisms $f, g \\colon a \\to b$ of $J$, there is a morphism $h \\colon b \\to c$ with $h \\circ f = h \\circ g$.\n\n## Remarks\n\nEquivalently, every diagram $F \\colon K \\to J$ with $K$ a finite category has a cocone.\nA colimit over a filtered category is a *filtered colimit*.\nThe dual notion is a *cofiltered* category.\n\nExample: a [[directed set|D9RY4667K6]] $(I, \\le)$, seen as a category with one arrow $i \\to j$ when $i \\le j$, is filtered.\nClause 1 holds because a directed set is not empty, and clause 2 is the existence of a common upper bound.\nClause 3 holds because there is at most one arrow between two objects, so $f = g$.\n\nNon-example: the category with two objects $a, b$, two arrows $f, g \\colon a \\to b$ and only identities besides, is not filtered.\nClause 3 fails: the only arrow out of $b$ is $\\id_b$, and $\\id_b \\circ f \\neq \\id_b \\circ g$.\n",
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