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

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

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:

Examples

There is a multicategory whose objects are (small) sets, where a morphism from the sets X1, X2, ..., and Xn to the set Y is an n-ary function, that is a function from the Cartesian product X1 × X2 × ... × Xn 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 Xn to the vector space Y is a multilinear operator, that is a linear transformation from the tensor product X1X2 ⊗ ... ⊗ Xn 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 Xn to the C-object Y is a C-morphism from the monoidal product of X1, X2, ..., and Xn 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

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