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  "body": "\n## Summary\n\nA filtered category is a [[category|A2DE66937D]] in which every finite diagram has a cocone.\nIt generalizes a [[directed set|D9RY4667K6]]: any two objects map to a common object.\nAlso, any two parallel arrows become equal after composing with a further arrow.\nColimits over filtered categories behave like unions, and in sets they commute with finite limits.\nThey are the basis of the theory of locally presentable categories.\n\n## Explanations\n\n### L1 human\n\nThink of a process that runs forward without end, where any two stages are followed by a later stage that contains both.\nA filtered category is such a process, with the extra rule that two different ways of going forward can always be made to agree later on.\nA directed set, such as the natural numbers with their usual order, is the simplest example.\nA category with a terminal object is also filtered, since everything maps to the terminal object.\n",
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