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  "body": "\n## Summary\n\nA factorization system on a category is a pair $(E, M)$ of classes of morphisms.\nEvery morphism factors as a member of $E$ followed by a member of $M$, in an essentially unique way.\nThe two classes determine each other through a unique [[lifting property|AZK4RD8XV0]].\nThe standard example in sets is (surjections, injections), which factors a function through its image.\nA weaker variant, the weak factorization system, drops uniqueness and is a basic ingredient of model categories.\n\n## Explanations\n\n### L1 human\n\nEvery function factors as a surjection onto its image followed by the inclusion of the image.\nA factorization system abstracts this: two kinds of maps such that every map is first one kind, then the other.\nThe factorization is unique up to a unique isomorphism of the middle object.\nGroups have the same system with surjective and injective homomorphisms, and the middle object is the image subgroup.\n",
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