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  "body": "\n## Definition\n\n*Category theory* is the branch of mathematics whose objects of study are:\n\n1. [[categories|A2DE66937D]], made of objects, morphisms between objects, and an associative composition of morphisms with identities;\n2. [[functors|28NQ0KRHD3]], the maps between categories that preserve composition and identities;\n3. [[natural transformations|KYCPCC57FT]], the maps between two functors with the same source and target.\n\n## Remarks\n\nThe three levels are tied together: small categories and functors form a category, and functors from one category to another with natural transformations between them form a category again.\nExample: the statement \"a functor sends isomorphisms to isomorphisms\" belongs to category theory, since it only uses composition and identities.\nIndeed, if $g \\circ f$ and $f \\circ g$ are identities, then so are $F(g) \\circ F(f)$ and $F(f) \\circ F(g)$.\nNon-example: the statement \"every finite group of prime order is cyclic\" is not a categorical statement, since it refers to the elements of a group.\nIts main constructions are limits and colimits, adjoint functors, and the Yoneda lemma.\n",
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