multicategories

A multicategory is a generalization of a category in which morphisms can have several inputs. It consists of a class of objects, and for every finite sequence of objects A1, …, An and a target object B, a set of multimorphisms from (A1, …, An) to B. There are identity multimorphisms for unary inputs and a composition operation: given a multimorphism f: (A1, …, An) → B and, for each i, multimorphisms fi: (A_i1, …, A_i m(i)) → A_i, one can form a composite f ∘ (f1, …, fn): (A11, …, A1m(1), …, An1, …, Anm(n)) → B. These operations satisfy associativity and units, analogous to those in ordinary categories.

Two common variants are non-symmetric and symmetric multicategories. In a symmetric multicategory, there is a right action of the symmetric group on the inputs, so permuting the inputs yields equivalent multimorphisms. In a non-symmetric multicategory, such permutations are not allowed. Multicategories can be enriched over a monoidal category, yielding enriched multicategories with hom-sets replaced by objects of the enriching category.

Examples illustrate the concept. The multicategory of sets has objects as sets and multimorphisms from (A1, …, An) to B given by functions A1 × … × An → B. A vector space V with multimorphisms to W can be taken as multilinear maps from (V1, …, Vn) to W. A key relationship is that operads are special cases: an operad is a multicategory with a single object; more generally, a colored (or multi-object) operad is a multicategory with many objects.

Relation to other notions: every category is a multicategory where only unary multimorphisms exist, embedding Cat into Multicat. Conversely, a (colored) operad is a multicategory with restricted structure, and multifunctors between multicategories generalize ordinary functors. Multicategories provide a unifying framework for handling multi-input operations across algebra, topology, and higher category theory.