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:

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