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 category theory by Saunders Mac Lane, it has expanded more recently drawing upon 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 composed one with others.

[edit] Definition

In category theory, an operad is a multicategory with one object. More explicitly, 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

[edit] 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 $ψ$ , subject to the condition that

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

Another class is that of topological operad, where names such as little n-disks, little n-cubes, et cetera occur. 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.