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 (X₁, X₂, ..., X_n) of objects (for n := 0, 1, 2, ...) and object Y, a set of morphisms from X₁, X₂, ..., and X_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_1,1, X_1,2, ..., X_1,n1; X_2,1, X_2,2, ..., X_2,n2; ...; X_m,1, X_m,2, ..., X_m,nm) of objects, a sequence (Y₁, Y₂, ..., Y_m) of objects, and an object Z: if

• f₁ is a morphism from X_1,1, X_1,2, ..., and X_1,n to Y₁;
• f₂ is a morphism from X_2,1, X_2,2, ..., and X_2,n to Y₂;
• ...;
• f_m is a morphism from X_m,1, X_m,2, ..., and X_m,n to Y_m; and
• g is a morphism from Y₁, Y₂, ..., and Y_m to Z:

then there is a composite morphism g(f₁, f₂, ..., f_m) from X_1,1, X_1,2, ..., X_1,n1, X_2,1, X_2,2, ..., X_2,n2, ..., X_m,1, X_m,2, ..., and X_m,nm to Z. This must satisfy certain axioms:

• If m is 1, Z is Y, and g is the identity morphism for Y, then g(f) must equal f;
• if n₁ is 1, n₂ is 1, ..., n_m is 1, X₁ is Y₁, X₂ is Y₂, ..., X_m is Y_m, f₁ is the identity morphism for Y₁, f₂ is the identity morphism for Y₂, ..., and f_m is the identity morphism for Y_m, then g(f₁, f₂, ..., f_m) must equal g; and
• an associativity condition (involving a further level of composition) that takes a long time to write down.

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. [1])

References

• Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. http://www.maths.gla.ac.uk/~tl/book.html.