Operad Theory

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

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) 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'',

''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

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	CATEGORIES ABOUT OPERAD THEORY

	abstract algebra

	category theory

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Operad Theory