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  "body": "\n## Definition\n\nA *multicategory* $\\mathcal{M}$ consists of the following data.\n\n1. A class of objects.\n2. For each $n \\ge 0$, each list of objects $x_1, \\ldots, x_n$ and each object $y$, a set $\\mathcal{M}(x_1, \\ldots, x_n; y)$ of *multimorphisms* from $(x_1, \\ldots, x_n)$ to $y$.\n3. For each object $x$, an identity $1_x \\in \\mathcal{M}(x; x)$.\n4. Composition maps\n$$\\mathcal{M}(y_1, \\ldots, y_n; z) \\times \\prod_{i=1}^{n} \\mathcal{M}(x_{i1}, \\ldots, x_{ik_i}; y_i) \\to \\mathcal{M}(x_{11}, \\ldots, x_{1k_1}, \\ldots, x_{n1}, \\ldots, x_{nk_n}; z),$$\nwritten $(g, f_1, \\ldots, f_n) \\mapsto g \\circ (f_1, \\ldots, f_n)$.\n\nThese data satisfy two axioms.\n\n1. (associativity) $\\left(h \\circ (g_1, \\ldots, g_m)\\right) \\circ (\\mathbf{f}_1, \\ldots, \\mathbf{f}_m) = h \\circ \\left(g_1 \\circ \\mathbf{f}_1, \\ldots, g_m \\circ \\mathbf{f}_m\\right)$, where $\\mathbf{f}_i$ is the list of multimorphisms that are plugged into the inputs of $g_i$.\n2. (unit) $1_y \\circ (f) = f$ and $f \\circ (1_{x_1}, \\ldots, 1_{x_n}) = f$ for every $f \\in \\mathcal{M}(x_1, \\ldots, x_n; y)$.\n\n## Remarks\n\nExample: the objects are the [[vector spaces|000AYPA9AJ]] over a field $K$, and $\\mathcal{M}(V_1, \\ldots, V_n; W)$ is the set of [[multilinear maps|0443XNARDV]] $V_1 \\times \\cdots \\times V_n \\to W$; a map with $n = 0$ inputs is an element of $W$.\nComposition is substitution of maps into the arguments.\nA multilinear map with multilinear maps substituted in its arguments is again multilinear in the combined list of arguments, and substitution is associative, so the axioms hold.\n\nNon-example: keep the same objects but allow only maps with $n \\ge 2$ inputs.\nComposition is still defined, but there is no identity in $\\mathcal{M}(V; V)$, so the unit clause fails.\n\nA multicategory with exactly one object is a non-symmetric operad.\nA *symmetric* multicategory has in addition actions of the symmetric groups that permute the inputs, compatible with composition.\nRestricting to multimorphisms with exactly one input gives an ordinary category.\n",
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