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  "body": "\n## Summary\n\nA preadditive category is a [[category|A2DE66937D]] in which morphisms with the same source and target can be added, and composition distributes over this addition.\nEquivalently, it is a category [[enriched|BPH9SJ30Y0]] over abelian groups.\nModules over a ring, with their homomorphisms, are the basic example.\nPreadditive categories are the first step towards additive and abelian categories, the setting of homological algebra.\n\n## Explanations\n\n### L1 human\n\nTwo linear maps between the same vector spaces can be added, and the sum is again linear.\nMoreover $h \\circ (f + g) = h \\circ f + h \\circ g$, so composition and addition get along.\nA preadditive category asks for exactly this: every set of arrows from one object to another is an abelian group, and composing distributes over sums.\nA ring is the same thing as a preadditive category with one object, whose arrows are the elements of the ring and whose composition is multiplication.\n",
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