Radioactive Ion Beam Optimization

2013-03-25T06:41:34Z

99.225.8.60: /* Overview */

In [[mathematics]] (especially [[category theory]]), a '''multicategory''' is a generalization of the concept of [[category (math)|category]] that allows morphisms of multiple [[arity]]. If morphisms in a category are viewed as analogous to [[function (mathematics)|function]]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]] <math>(X_i)_{i \in n}</math> of objects (for von Neumann ordinal <math>n \in \mathbb{N}</math>) and object ''Y'', a set of ''morphisms'' from <math>(X_i)_{i \in n}</math> 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 <math>((X_{ij})_{i \in n_j})_{j \in m}</math> of objects, a sequence <math>(Y_i)_{i \in m}</math> of objects, and an object ''Z'': if
* for each <math>j \in m</math>, ''f''''j'' is a morphism from <math>(X_{ij})_{i \in n_j}</math> to ''Y''''j''; and
* ''g'' is a morphism from <math>(Y_i)_{i \in m}</math> to ''Z'':
then there is a composite morphism <math>g(f_i)_{i \in m}</math> from <math>(X_{ij})_{i \in n_j, j \in m}</math> 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 <math>i \in m</math>, ''n''''i'' = 1, <math>X_{0i} = Y_i</math>, and ''f''''i'' is the identity morphism for ''Y''''i'', then <math>g(f_i)_{i \in m} = g</math>; and
* an [[associativity]] condition: if for each <math>k \in m</math> and <math>j \in n_k</math>, <math>e_{jk}</math> is a morphism from <math>(W_{ijk})_{i \in o_{jk}}</math> to <math>X_{jk}</math>, then <math>g\left(f_j(e_{ij})_{i \in n_j}\right)_{j \in m} = g(f_i)_{i \in m}(e_{ij})_{i \in n_j, j \in m}</math> are identical morphisms from <math>(W_{ijk})_{i \in o_{jk}, j \in n_k, k \in m}</math> to ''Z''.

== Examples ==

There is a multicategory whose objects are (small) [[set (math)|set]]s, where a morphism from the sets ''X''1, ''X''2, ..., and ''X''''n'' to the set ''Y'' is an [[binary function|''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 space]]s (over the [[rational number]]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]], 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. (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])

== References ==
*{{cite book
| author = Tom Leinster
| year = 2004
| title = Higher Operads, Higher Categories
| publisher = Cambridge University Press
| url = http://www.maths.gla.ac.uk/~tl/book.html
}}

[[Category:Category theory]]

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Radioactive Ion Beam Optimization