Multicategory

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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[edit]

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, fj is a morphism from (X_{ij})_{i \in n_j} to Yj; 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, ni = 1, X_{0i} = Y_i, and fi is the identity morphism for Yi, 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[edit]

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. [1])

References[edit]