Multicategory

In mathematics (especially category theory), a multicategory is a generalization of the concept of category that allows morphisms of multiple arity. If morphisms in a category are viewed as analogous to functions, then morphisms in a multicategory are analogous to functions of several variables.

Definition

A multicategory consists of

a collection (often a proper class) of objects;
for every finite sequence $(X_i)_{i \in n}$ of objects (for von Neumann ordinal $n \in \mathbb{N}$ ) and object Y, a set of morphisms from $(X_i)_{i \in n}$ to Y; and
for every object X, a special identity morphism (with n = 1) from X to X.

Additionally, there are composition operations: Given a sequence of sequences $((X_{ij})_{i \in n_j})_{j \in m}$ of objects, a sequence $(Y_i)_{i \in m}$ of objects, and an object Z: if

for each $j \in m$ , f_j is a morphism from $(X_{ij})_{i \in n_j}$ to Y_j; and
g is a morphism from $(Y_i)_{i \in m}$ to Z:

then there is a composite morphism $g(f_i)_{i \in m}$ from $(X_{ij})_{i \in n_j, j \in m}$ to Z. This must satisfy certain axioms:

If m = 1, Z = Y₀, and g is the identity morphism for Y₀, then g(f₀) = f₀;
if for each $i \in m$ , n_i = 1, $X_{0i} = Y_i$ , and f_i is the identity morphism for Y_i, then $g(f_i)_{i \in m} = g$ ; and
an associativity condition: if for each $k \in m$ and $j \in n_k$ , $e_{jk}$ is a morphism from $(W_{ijk})_{i \in o_{jk}}$ to $X_{jk}$ , then $g\left(f_j(e_{ij})_{i \in n_j}\right)_{j \in m} = g(f_i)_{i \in m}(e_{ij})_{i \in n_j, j \in m}$ are identical morphisms from $(W_{ijk})_{i \in o_{jk}, j \in n_k, k \in m}$ to Z.

Examples

There is a multicategory whose objects are (small) sets, where a morphism from the sets X₁, X₂, ..., and X_n to the set Y is an n-ary function, that is a function from the Cartesian product X₁ × X₂ × ... × X_n to Y.

There is a multicategory whose objects are vector spaces (over the rational numbers, say), where a morphism from the vector spaces X₁, X₂, ..., and X_n to the vector space Y is a multilinear operator, that is a linear transformation from the tensor product X₁ ⊗ X₂ ⊗ ... ⊗ X_n to Y.

More generally, given any monoidal category C, there is a multicategory whose objects are objects of C, where a morphism from the C-objects X₁, X₂, ..., and X_n to the C-object Y is a C-morphism from the monoidal product of X₁, X₂, ..., and X_n to Y.

An operad is a multicategory with one unique object; except in degenerate cases, such a multicategory does not come from a monoidal category. (The term "operad" is often reserved for symmetric multicategories; terminology varies. )

References

Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press.