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. Multicategories are also sometimes called operads, or colored operads.
Definition.
A (non-symmetric) multicategory consists of
Additionally, there are composition operations: Given a sequence of sequences formula_4 of objects, a sequence formula_5 of objects, and an object "Z": if
then there is a composite morphism formula_9 from formula_10 to "Z". This must satisfy certain axioms:
Comcategories.
A "comcategory" (co-multi-category) is a totally ordered set "O" of objects, a set "A" of "multiarrows" with two functions
formula_21
formula_22
where "O"% is the set of all finite ordered sequences of elements of "O". The dual image of a multiarrow "f" may be summarized
formula_23
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.
A "multiorder" is a comcategory satisfying the following conditions.
Multiorders are a generalization of partial orders (posets), and were first introduced (in passing) by Tom Leinster.
Examples.
There is a multicategory whose objects are (small) sets, where a morphism from the sets "X"1, "X"2, ..., and "X""n" to the set "Y" is an "n"-ary function,
that is a function from the Cartesian product "X"1 × "X"2 × ... × "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 "X"1, "X"2, ..., and "X""n" to the vector space "Y" is a multilinear operator, that is a linear transformation from the tensor product "X"1 ⊗ "X"2 ⊗ ... ⊗ "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 "X"1, "X"2, ..., and "X""n" to the C-object "Y" is a C-morphism from the monoidal product of "X"1, "X"2, ..., 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.
Examples of multiorders include "pointed multisets" , "integer partitions" , and "combinatory separations" . The triangles (or compositions) of any multiorder are morphisms of a (not necessarily associative) category of "contractions" and a comcategory of "decompositions". The contraction category for the multiorder of "multimin partitions" is the simplest known category of multisets.
Applications.
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.
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.
History.
Multicategories were first introduced under that name by Jim Lambek in "Deductive systems and categories II" (1969) 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".