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.
A multicategory consists of
- a collection (often a proper class) of objects;
- for every finite sequence of objects (for von Neumann ordinal ) and object Y, a set of morphisms from 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 of objects, a sequence of objects, and an object Z: if
- for each , fj is a morphism from to Yj; and
- g is a morphism from to Z:
then there is a composite morphism from 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 , ni = 1, , and fi is the identity morphism for Yi, then ; and
- an associativity condition: if for each and , is a morphism from to , then are identical morphisms from to Z.
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 X1 ⊗ X2 ⊗ ... ⊗ 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. )
- Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press.