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  "body": "\n## Definition\n\nA *category* $\\mathcal{C}$ consists of the following data:\n\n1. a class $\\mathrm{Ob}(\\mathcal{C})$, whose members are called the objects of $\\mathcal{C}$;\n2. for each pair of objects $A, B$, a class $\\Hom_{\\mathcal{C}}(A, B)$ of morphisms from $A$ to $B$, written $f \\colon A \\to B$, with these classes pairwise disjoint;\n3. for each triple of objects $A, B, C$, a composition map $\\Hom_{\\mathcal{C}}(B, C) \\times \\Hom_{\\mathcal{C}}(A, B) \\to \\Hom_{\\mathcal{C}}(A, C)$, written $(g, f) \\mapsto g \\circ f$;\n4. for each object $A$, an identity morphism $1_A \\in \\Hom_{\\mathcal{C}}(A, A)$.\n\nThese data must satisfy two axioms:\n\n1. associativity: $h \\circ (g \\circ f) = (h \\circ g) \\circ f$ for all $f \\colon A \\to B$, $g \\colon B \\to C$ and $h \\colon C \\to D$;\n2. identity: $f \\circ 1_A = f = 1_B \\circ f$ for all $f \\colon A \\to B$.\n\n## Remarks\n\nThe category is *locally small* when every $\\Hom_{\\mathcal{C}}(A, B)$ is a [[set|0BSJ7RQ9BW]], and *small* when moreover $\\mathrm{Ob}(\\mathcal{C})$ is a set.\nExample: $\\mathbf{Set}$ has sets as objects and [[functions|0EPAT51DXQ]] $A \\to B$ as morphisms, with the usual composition and the identity functions.\nComposition of functions is associative and the identity functions are neutral, so both axioms hold.\nNon-example: take sets as objects and, as morphisms $A \\to B$, the functions $A \\to B$ whose image has at most one point.\nA composite of two such functions again has at most one point in its image, so composition is defined.\nThe identity function of a set with two elements is not such a morphism, so clause 4 fails.\nMany authors write $\\mathcal{C}(A, B)$ or $\\hom(A, B)$ for $\\Hom_{\\mathcal{C}}(A, B)$, and $gf$ for $g \\circ f$.\n",
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