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 $f \in \mathcal{M}(x_1, \ldots, x_n; y)$.