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  "body": "\n## Definition\n\nLet $\\mathbb{B}$ be the category whose objects are the [[finite sets|0BBTH3YSZA]] and whose morphisms are the bijections.\nA *combinatorial species* is a functor $F \\colon \\mathbb{B} \\to \\mathbb{B}$.\nThus $F$ assigns:\n\n1. to each finite set $U$ a finite set $F[U]$, whose elements are the *$F$-structures* on $U$;\n2. to each bijection $\\sigma \\colon U \\to V$ a bijection $F[\\sigma] \\colon F[U] \\to F[V]$, the *transport* along $\\sigma$;\n3. in such a way that $F[\\id_U] = \\id_{F[U]}$ and $F[\\tau \\circ \\sigma] = F[\\tau] \\circ F[\\sigma]$ for bijections $\\sigma \\colon U \\to V$ and $\\tau \\colon V \\to W$.\n\n## Remarks\n\nThe *exponential generating function* of $F$ is $F(x) = \\sum_{n \\ge 0} \\abs{F[\\set{1, \\ldots, n}]}\\, \\frac{x^n}{n!}$.\n\nExample: the species $L$ of linear orders assigns to $U$ the set of total orders on $U$.\nA bijection $\\sigma$ transports an order $u_1 \u003c \\cdots \u003c u_n$ to $\\sigma(u_1) \u003c \\cdots \u003c \\sigma(u_n)$, and the functor rules hold.\nSince $\\abs{L[\\set{1, \\ldots, n}]} = n!$, its generating function is $\\frac{1}{1 - x}$.\nThe species $E$ of sets, with $E[U] = \\set{U}$, has generating function $e^x$.\n\nNon-example: let $F[U] = \\set{U}$ if $U \\subseteq \\N$ and $F[U] = \\emptyset$ otherwise.\nFor a bijection $\\sigma$ from $\\set{1}$ to $\\set{a}$ with $a \\notin \\N$, clause 2 needs a bijection from a one-element set to the empty set, which does not exist.\nSo $F$ is not a species.\n\nSum, product, composition and derivative of species correspond to the same operations on exponential generating functions.\nSome authors let $F$ take values in the category of finite sets and all functions; this gives the same notion, since a functor sends bijections to bijections.\n",
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