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  "body": "\n## Summary\n\nCategorical logic is the branch of mathematics that studies logic with the tools of [[category theory|Q1RZT7DV0F]].\nA logical theory is matched with a category built from its syntax.\nA model of the theory is then a structure-preserving [[functor|28NQ0KRHD3]] from that category to another category, such as the category of sets.\nTypical correspondences match the simply typed [[lambda calculus|BD756KR8FM]] with [[Cartesian closed categories|BFBNQ36QS8]], and higher-order [[intuitionistic logic|Q11N8X99FK]] with [[toposes|6837RS7N82]].\nThese correspondences let one transfer results between logic, algebra, geometry and the semantics of programming languages.\n\n## Explanations\n\n### L1 human\n\nIn ordinary logic one writes formulas and proofs, and then interprets them in sets.\nCategorical logic allows the interpretation to take place in any suitable category, not only in sets.\nTypes become objects, terms become morphisms, and logical connectives become operations such as products and exponentials.\nFor example, a proof that $A$ and $B$ together give $C$ becomes a morphism $A \\times B \\to C$, and the rule \"from $A$ and $B$ infer $A$\" becomes the projection $A \\times B \\to A$.\n",
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