If morphisms in a category are viewed as analogous to functions, then morphisms in a multicategory are analogous to functions of several variables.
where O% is the set of all finite ordered sequences of elements of O.
A comcategory C also has a multiproduct with the usual character of a composition operation.
C is said to be associative if there holds a multiproduct axiom in relation to this operator.
Any multicategory, symmetric or non-symmetric, together with a total-ordering of the object set, can be made into an equivalent comcategory.
Multiorders are a generalization of partial orders (posets), and were first introduced (in passing) by Tom Leinster.
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.
The triangles (or compositions) of any multiorder are morphisms of a (not necessarily associative) category of contractions and a comcategory of decompositions.
[2] Multicategories are often incorrectly considered to belong to higher category theory, as their original application was the observation that the operators and identities satisfied by higher categories are the objects and multiarrows of a multicategory.
The study of n-categories was in turn motivated by applications in algebraic topology and attempts to describe the homotopy theory of higher dimensional manifolds.
However it has mostly grown out of this motivation and is now also considered to be part of pure mathematics.
[1] The correspondence between contractions and decompositions of triangles in a multiorder allows one to construct an associative algebra called its incidence algebra.
Any element that is nonzero on all unit arrows has a compositional inverse, and the Möbius function of a multiorder is defined as the compositional inverse of the zeta function (constant-one) in its incidence algebra.
Multicategories were first introduced under that name by Jim Lambek in "Deductive systems and categories II" (1969)[3] He mentions (p. 108) that he was "told that multicategories have also been studied by [Jean] Benabou and [Pierre] Cartier", and indeed Leinster opines that "the idea might have occurred to anyone who knew what both a category and a multilinear map were".