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  "body": "\n## Definition\n\nLet $(\\mathcal{C}, \\otimes, I, \\alpha, \\lambda, \\rho)$ be a [[monoidal category|M3MSQ3NKAE]].\nA *monoid object* in $\\mathcal{C}$ is an object $M$ together with morphisms $\\mu \\colon M \\otimes M \\to M$ (multiplication) and $\\eta \\colon I \\to M$ (unit) such that:\n\n1. associativity: $\\mu \\circ (\\mu \\otimes \\id_M) = \\mu \\circ (\\id_M \\otimes \\mu) \\circ \\alpha_{M,M,M}$ as morphisms $(M \\otimes M) \\otimes M \\to M$;\n2. left unit: $\\mu \\circ (\\eta \\otimes \\id_M) = \\lambda_M$ as morphisms $I \\otimes M \\to M$;\n3. right unit: $\\mu \\circ (\\id_M \\otimes \\eta) = \\rho_M$ as morphisms $M \\otimes I \\to M$.\n\nA morphism of monoid objects $(M, \\mu, \\eta) \\to (M', \\mu', \\eta')$ is a morphism $f \\colon M \\to M'$ with $f \\circ \\mu = \\mu' \\circ (f \\otimes f)$ and $f \\circ \\eta = \\eta'$.\n\n## Remarks\n\nExample: in $(\\mathbf{Set}, \\times, \\set{*})$, a monoid object is a set $M$ with $\\mu(a, b) = ab$ and $\\eta(*) = e$; the three axioms say $(ab)c = a(bc)$, $ea = a$ and $ae = a$, so it is an ordinary [[monoid|0N6R902FG1]].\nExample: in $(\\mathbf{Ab}, \\otimes, \\Z)$, a monoid object is a ring, since a bilinear multiplication is a morphism $R \\otimes R \\to R$.\nNon-example: in $(\\mathbf{Set}, \\times, \\set{*})$, the set $\\Z$ with $\\mu(a, b) = a - b$ and $\\eta(*) = 0$ fails associativity, because $(1 - 1) - 1 = -1$ while $1 - (1 - 1) = 1$.\nA monoid object in a symmetric monoidal category is *commutative* if $\\mu \\circ \\gamma_{M,M} = \\mu$, where $\\gamma$ is the symmetry.\n",
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