Multicategory
Encyclopedia
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 function
s, then morphisms in a multicategory are analogous to functions of several variables.
Additionally, there are composition operations: Given a sequence of sequences (X1,1, X1,2, ..., X1,n1; X2,1, X2,2, ..., X2,n2; ...; Xm,1, Xm,2, ..., Xm,nm) of objects, a sequence (Y1, Y2, ..., Ym) of objects, and an object Z: if
then there is a composite morphism g(f1, f2, ..., fm) from X1,1, X1,2, ..., X1,n1, X2,1, X2,2, ..., X2,n2, ..., Xm,1, Xm,2, ..., and Xm,nm to Z. This must satisfy certain axioms:
,
that is a function from the Cartesian product
X1 × X2 × ... × Xn to Y.
There is a multicategory whose objects are vector space
s (over the rational number
s, 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.
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. http://golem.ph.utexas.edu/category/2006/09/this_weeks_finds_in_mathematic.html#c004579)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
(especially category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
), a multicategory is a generalization of the concept of category that allows morphisms of multiple arity
Arity
In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
. If morphisms in a category are viewed as analogous to function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s, then morphisms in a multicategory are analogous to functions of several variables.
Definition
A multicategory consists of- a collection (often a proper class) of objects;
- for every finite sequence (X1, X2, ..., Xn) of objects (for n := 0, 1, 2, ...) and object Y, a set of morphisms from X1, X2, ..., and Xn 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 (X1,1, X1,2, ..., X1,n1; X2,1, X2,2, ..., X2,n2; ...; Xm,1, Xm,2, ..., Xm,nm) of objects, a sequence (Y1, Y2, ..., Ym) of objects, and an object Z: if
- f1 is a morphism from X1,1, X1,2, ..., and X1,n to Y1;
- f2 is a morphism from X2,1, X2,2, ..., and X2,n to Y2;
- ...;
- fm is a morphism from Xm,1, Xm,2, ..., and Xm,n to Ym; and
- g is a morphism from Y1, Y2, ..., and Ym to Z:
then there is a composite morphism g(f1, f2, ..., fm) from X1,1, X1,2, ..., X1,n1, X2,1, X2,2, ..., X2,n2, ..., Xm,1, Xm,2, ..., and Xm,nm to Z. This must satisfy certain axioms:
- If m is 1, Z is Y, and g is the identity morphism for Y, then g(f) must equal f;
- if n1 is 1, n2 is 1, ..., nm is 1, X1 is Y1, X2 is Y2, ..., Xm is Ym, f1 is the identity morphism for Y1, f2 is the identity morphism for Y2, ..., and fm is the identity morphism for Ym, then g(f1, f2, ..., fm) must equal g; and
- an associativityAssociativityIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
condition (involving a further level of composition) that takes a long time to write down.
Examples
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 functionBinary function
In mathematics, a binary function, or function of two variables, is a function which takes two inputs.Precisely stated, a function f is binary if there exists sets X, Y, Z such that\,f \colon X \times Y \rightarrow Z...
,
that is a function from the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
X1 × X2 × ... × Xn to Y.
There is a multicategory whose objects are vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s (over the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, 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
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
from the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
X1 ⊗ X2 ⊗ ... ⊗ Xn to Y.
More generally, given any monoidal category
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
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. http://golem.ph.utexas.edu/category/2006/09/this_weeks_finds_in_mathematic.html#c004579)