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  "body": "\n## Definition\n\nLet $\\aleph_0$ be the [[category|A2DE66937D]] whose objects are the natural numbers $n \\ge 0$.\nEach $n$ is viewed as the set $\\set{1, \\ldots, n}$, and the morphisms are all functions between these sets.\nA *Lawvere theory* is a small category $\\mathcal{T}$ with finite [[products|S5744QZV4N]], together with a [[functor|28NQ0KRHD3]] $I \\colon \\aleph_0^{\\mathrm{op}} \\to \\mathcal{T}$ that is bijective on objects and preserves finite products.\nA *model* of $\\mathcal{T}$ is a functor $\\mathcal{T} \\to \\mathbf{Set}$ that preserves finite products.\n\n## Remarks\n\nIn $\\aleph_0^{\\mathrm{op}}$, the object $n$ is the product of $n$ copies of $1$, with projections given by the inclusions $\\set{1} \\to \\set{1, \\ldots, n}$.\nSo in $\\mathcal{T}$ every object is a power $1^n$ of the object $1$.\nA model $M$ is determined by the set $M(1)$ and the maps $M(1)^n \\to M(1)$ given by the morphisms $n \\to 1$.\n\nExample: for a [[variety|AJNZJD8442]] $\\mathcal{V}$ with free algebras $F(n)$ on $n$ generators, let $\\mathcal{T}_{\\mathcal{V}}(n, m)$ be the set of homomorphisms $F(m) \\to F(n)$, composed in the opposite order.\nThis is a Lawvere theory.\nHere $\\mathcal{T}_{\\mathcal{V}}(n, 1) \\cong F(n)$ is the set of $n$-ary terms modulo the identities of $\\mathcal{V}$.\nThe object $n$ is the product of $n$ copies of $1$ because $F(n)$ is the coproduct of $n$ copies of $F(1)$ in $\\mathcal{V}$.\nIts category of models is equivalent to $\\mathcal{V}$.\n\nNon-example: the category $\\aleph_0$ itself, not its opposite, has finite products, given by cartesian products, so the product of $n$ and $m$ is $n m$.\nIt is not a Lawvere theory, since in a Lawvere theory every object is a power of one object $x$.\nHere $2$ and $3$ are not both powers of the same natural number.\n",
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