In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
(especially
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
), a multicategory is a generalization of the concept of
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
that allows morphisms of multiple
arity
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In ...
. If morphisms in a category are viewed as analogous to
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s, then morphisms in a multicategory are analogous to functions of several variables. Multicategories are also sometimes called
operads
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defin ...
, or colored operads.
Definition
A (non-symmetric) multicategory consists of
* a collection (often a
proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
) of ''objects'';
* for every
finite sequence
In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
of objects (for von Neumann ordinal
) and object ''Y'', a set of ''morphisms'' from
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
of objects, a sequence
of objects, and an object ''Z'': if
* for each
, ''f''
''j'' is a morphism from
to ''Y''
''j''; and
* ''g'' is a morphism from
to ''Z'':
then there is a composite morphism
from
to ''Z''. This must satisfy certain axioms:
* If ''m'' = 1, ''Z'' = ''Y''
0, and ''g'' is the identity morphism for ''Y''
0, then ''g''(''f''
0) = ''f''
0;
* if for each
, ''n''
''j'' = 1,
, and ''f''
''j'' is the identity morphism for ''Y''
''j'', then
; and
* an
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
condition: if for each
and
,
is a morphism from
to
, then
are identical morphisms from
to ''Z''.
Comcategories
A ''comcategory'' (co-multi-category) is a
totally ordered set
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
''O'' of objects, a set ''A'' of ''multiarrows'' with two functions
where ''O''
% is the set of all finite ordered sequences of elements of ''O''. The dual image of a multiarrow ''f'' may be summarized
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.
* There is at most one multiarrow with given head and ground.
* Each object ''x'' has a unit multiarrow.
* A multiarrow is a unit if its ground has one entry.
Multiorders are a generalization of partial orders (posets), and were first introduced (in passing) by Tom Leinster.
[, Example 2.1.7, page 37]
Examples
There is a multicategory whose objects are (small)
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s, 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
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
''X''
1 × ''X''
2 × ... × ''X''
''n'' to ''Y''.
There is a multicategory whose objects are
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s (over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, say), where a morphism from the vector spaces ''X''
1, ''X''
2, ..., and ''X''
''n'' to the vector space ''Y'' is a
multilinear operator
In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
, that is a
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
from the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
''X''
1 ⊗ ''X''
2 ⊗ ... ⊗ ''X''
''n'' to ''Y''.
More generally, given any
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
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
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...
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 In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher cate ...
, 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
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and attempts to describe the
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
of higher dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s. However it has mostly grown out of this motivation and is now also considered to be part of pure mathematic
The correspondence between contractions and decompositions of triangles in a multiorder allows one to construct an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
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
Benabou">eanBenabou and
ierreCartier", and indeed Leinster opines that "the idea might have occurred to anyone who knew what both a category and a multilinear map were".
References
*{{cite journal, last1=Garner, first1=Richard, title=Polycategories via pseudo-distributive laws, journal=
Advances in Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes.
At the origin, the journal aimed ...
, date=2008, volume=218, issue=3, pages=781–827, doi=10.1016/j.aim.2008.02.001, doi-access=free, arxiv=math/0606735, s2cid=17057235
Category theory