Operad theory

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Operad theory is a field of abstract algebra concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. Originating from work in algebraic topology by J. Peter May, Boardman, and Vogt, it has more recently found many applications, drawing for example on work by Maxim 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.

1 Definition
2 Examples
3 Origins of the term
4 See also
5 References

[edit] Definition

In category theory, an operad without permutations (sometimes called a non-symmetric, non- $Σ$ or plain operad) 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

$\begin{matrix} P(n)\times P(k_1)\times\cdots\times P(k_n)&\to&P(k_1+\cdots+k_n)\\ (\theta,\theta_1,\ldots,\theta_n)&\mapsto&\theta\circ(\theta_1,\ldots,\theta_n) \end{matrix}$

called composition,

an element $1$ in $P (1)$ called the identity,

satisfying the following coherence properties

associativity:

$\theta\circ(\theta_1\circ(\theta_{1,1},\ldots,\theta_{1,k_1}),\ldots,\theta_n\circ(\theta_{n,1},\ldots,\theta_{n,k_n})) = (\theta\circ(\theta_1,\ldots,\theta_n))\circ(\theta_{1,1},\ldots,\theta_{1,k_1},\ldots,\theta_{n,1},\ldots,\theta_{n,k_n})$

identity:

$\theta\circ(1,\ldots,1)=\theta=1\circ\theta$ (where the number of arguments correspond to the arities of the operations).

A morphism of operads $f:P\to Q$ consists of a sequence

$(f_n:P(n)\to Q(n))_{n\in\mathbb{N}}$

which

preserves composition: for every n-ary operation $θ$ and operations $θ 1$ , ..., $θ n$ ,

$f(\theta\circ(\theta_1,\ldots,\theta_n)) = f(\theta)\circ(f(\theta_1),\ldots,f(\theta_n))$

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 $θ n$ . The permutation actions are additional structure that is vital to the original and most later applications.

[edit] Examples

Operadic composition in the little 2-discs operad.

Operadic composition in the operad of symmetries.

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 binary operations.

Thus, the associative operad is generated by a binary operation $ψ$ , subject to the condition that

$\psi\circ(\psi,1)=\psi\circ(1,\psi).$

This operad is terminal in the category of non-symmetric operads, as it has exactly one n-ary operation for each n. For this reason, it is sometimes written as 1 by category theorists (by analogy with the one-point set, which is terminal in the category of sets). The terminal symmetric operad is the operad whose algebras are commutative monoids, which also has one n-ary operation for each n, with each $S n$ acting trivially.

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.

There is an operad for which each $P (n)$ is given by the symmetric group $S n$ . The composite $\sigma \circ (\tau_1, \dots, \tau_n)$ permutes its inputs in blocks according to $σ$ , and within blocks according to the appropriate $τ i$ . Similarly, there is an operad for which each $P (n)$ is given by the Artin braid group $B n$ .

[edit] Origins of the term

The word "operad" was also created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). 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)

[edit] See also

PRO (category theory)

[edit] References

Boardman, J. M. & Vogt, R. M. (1973), Homotopy Invariant Algebraic Structures on Topological Spaces, vol. 347, Lecture Notes in Mathematics, Springer-Verlag, ISBN 3540064796 .