Operad Theory Website Links For
Theory 		 

Information About

Operad Theory
  CATEGORIES ABOUT OPERAD THEORY
   
abstract algebra
   
category theory
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



many other applications, drawing for example on work by Kontsevich on graph homology.

An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others.


DEFINITION

In Category Theory , an operad without permutations is a Multicategory with one object. More explicitly, such an operad consists of
  • a sequence (P(n))_{n\in\mathbb{N}} of sets, whose elements are called ''n-ary operations'',

  • for each integers n, k_1, ..., k_n a function

    • :

\begin{matrix}
P(n) imes P(k_1) imes\cdots imes P(k_n)& o&P(k_1+\cdots+k_n)\
( heta, heta_1,\ldots, heta_n)&\mapsto& heta\circ( heta_1,\ldots, heta_n)
\end{matrix}

called ''composition'',
  • an element 1 in P(1) called the ''identity'',

    • satisfying the following coherence properties

  • ''associativity'':

    • :

heta\circ( heta_1\circ( heta_{1,1},\ldots, heta_{1,k_1}),\ldots, heta_n\circ( heta_{n,1},\ldots, heta_{n,k_n}))
=
( heta\circ( heta_1,\ldots, heta_n))\circ( heta_{1,1},\ldots, heta_{1,k_1},\ldots, heta_{n,1},\ldots, heta_{n,k_n})

  • ''identity'':

    • : heta\circ(1,\ldots,1)= heta=1\circ heta

(where the number of arguments correspond to the arities of the operations).

A morphism of operads f:P o Q consists of a sequence
:(f_n:P(n) o Q(n))_{n\in\mathbb{N}}
which
  • preserves composition: for every ''n''-ary operation heta and operations heta_1, ..., heta_n,

    • :

f( heta\circ( heta_1,\ldots, heta_n))
=
f( heta)\circ(f( heta_1),\ldots,f( heta_n))

  • preserves identity:

    • :f(1)=1.


Operads were originally defined topologically, by May, but his full definition requires symmetric
group actions on the P(n) that are suitably related to the maps heta_n. The permutation actions
are additional structure that is vital to the original and most later applications.


EXAMPLES


One class of examples of operads are those capturing the structures of algebraic structures, such as associative algebras, commutative algebras and Lie algebras. Each of these can be exhibited as a finitely presented operad, in each of these three generated by 2-ary operations.

Thus, the associative operad is generated by a 2-ary operation \psi, subject to the condition that
:\psi\circ(\psi,1)=\psi\circ(1,\psi).

In many examples the P(n) are not just sets but rather topological spaces. Some names of important
examples are the ''little n-disks'', ''little n-cubes'', and ``linear isometries'' operads. The idea behind the
little n-disks operad comes from homotopy theory, and the idea is that an element of P(n)
is an arrangement of ''n'' disks within the unit disk. Now, the identity is the unit disk as a subdisk of itself, and composition of arrangements is by scaling the unit disk down into the disk that corresponds to the slot in the composition, and inserting the scaled contents there.


ORIGINS OF THE TERM

The word "operad" was also created by May. Regarding its creation, he wrote: "The name 'operad' is a word that I coined myself, spending a week thinking of nothing else." (http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf Page 2)


REFERENCES