---
tag: 97ZY7H816E
kind: definition
name: Multicategory
msc: [18M65, 18M60]
defines: MFB1Q9JTPD
uses: [A2DE66937D, 0443XNARDV, 000AYPA9AJ]
sources:
  - url: https://en.wikipedia.org/wiki/Multicategory
---

## Definition

A *multicategory* $\mathcal{M}$ consists of the following data.

1. A class of objects.
2. 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 $y$.
3. For each object $x$, an identity $1_x \in \mathcal{M}(x; x)$.
4. Composition maps
$$\mathcal{M}(y_1, \ldots, y_n; z) \times \prod_{i=1}^{n} \mathcal{M}(x_{i1}, \ldots, x_{ik_i}; y_i) \to \mathcal{M}(x_{11}, \ldots, x_{1k_1}, \ldots, x_{n1}, \ldots, x_{nk_n}; z),$$
written $(g, f_1, \ldots, f_n) \mapsto g \circ (f_1, \ldots, f_n)$.

These data satisfy two axioms.

1. (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 $g_i$.
2. (unit) $1_y \circ (f) = f$ and $f \circ (1_{x_1}, \ldots, 1_{x_n}) = f$ for every $f \in \mathcal{M}(x_1, \ldots, x_n; y)$.

## Remarks

Example: 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 $W$.
Composition is substitution of maps into the arguments.
A 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.

Non-example: keep the same objects but allow only maps with $n \ge 2$ inputs.
Composition is still defined, but there is no identity in $\mathcal{M}(V; V)$, so the unit clause fails.

A multicategory with exactly one object is a non-symmetric operad.
A *symmetric* multicategory has in addition actions of the symmetric groups that permute the inputs, compatible with composition.
Restricting to multimorphisms with exactly one input gives an ordinary category.