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. Multicategories are also sometimes called operads, or colored operads.
Definition A (non-symmetric) multicategory consists of
a collection (often a proper class) of objects; for every finite sequence ( X i ) i ∈ [ n ] {\displaystyle (X_{i})_{i\in [n]}} of objects ( [ n ] = { 0 , 1 , 2 , . . . , n } {\displaystyle [n]=\{0,1,2,...,n\}} ) and every object Y, a set of morphisms from ( X i ) i ∈ n {\displaystyle (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 i j ) i ∈ n j ) j ∈ m {\displaystyle ((X_{ij})_{i\in n_{j}})_{j\in m}} of objects, a sequence ( Y j ) j ∈ m {\displaystyle (Y_{j})_{j\in m}} of objects, and an object Z: if
for each j ∈ m {\displaystyle j\in m} , fj is a morphism from ( X i j ) i ∈ n j {\displaystyle (X_{ij})_{i\in n_{j}}} to Yj; and g is a morphism from ( Y j ) j ∈ m {\displaystyle (Y_{j})_{j\in m}} to Z: then there is a composite morphism g ( f j ) j ∈ m {\displaystyle g(f_{j})_{j\in m}} from ( X i j ) i ∈ n j , j ∈ m {\displaystyle (X_{ij})_{i\in n_{j},j\in m}} to Z. This must satisfy certain axioms:
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