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\centerline {\bf Operad}
\medskip

An {\it algebraic operad} is an algebraic device to encode a type of algebra. It is
analogous, for algebras, to the table of mutiplication for groups. Starting with a type of
algebras ${\cal P}$ (for instance associative algebras $As$, Lie algebras $Lie$,
Poisson algebras $Pois$, or algebras up to homotopy, etc.), we consider the free algebra ${\cal P}(V)$ over the
vector space $V$. By the Schur lemma, over the rationals it is of the form 
$$
{\cal P}(V) = \bigoplus_{n\geq 0} {\cal P}(n)\otimes_{S_n}V^{\otimes n}\ ,
$$
where $S_n$ is the symmetric group and ${\cal P}(n)$ some right $S_n$-module. The
space ${\cal P}(n)$ is the space of operations on $n$ variables: for any ${\cal
P}$-algebra $A$ there is a map ${\cal P}(n)\otimes A^{\otimes n}\to A$. 

Let $A$ be a ${\cal
P}$-algebra and $f:W\to A$ a linear map. It extends to a ${\cal P}$-algebra map ${\cal P}(W) \to
A$. Taking
$W=A={\cal P}(V)$ and $f={\rm Id}$ gives a functorial map ${\cal P}({\cal P}(V))\to {\cal P}(V)$.
This transformation of functor $\mu : {\cal P}\circ {\cal P}\to {\cal P}$ is clearly associative.
The pair $({\cal P}, \mu)$ is called an {\it algebraic operad}.
\smallskip
\noindent {\bf Examples}.

\noindent 1. For the operad $Com$ of (nonunital) associative and commutative algebras, the
space $Com(n)$ is the trivial representation of $S_n$ of dimension 1.

\noindent 2. For the operad $Lie$ of Lie algebras, the space
$Lie(n)$ is of dimension $(n-1)!$, its structure is complicated to describe.

\noindent 3. For the operads $As$ of associative algebras,
$Leib$ of Leibniz algebras, $Pois$ of Poisson algebras, the spaces
$As(n)$, $Leib(n)$, $Pois(n)$ are the regular representation of $S_n$.
So the operads $As, Leib, Pois$ differ only by their composition maps.

\noindent 4. For the operad $Dias$ of dialgebras, the space
$Dias(n)$ is the sum of $n$ copies of the regular representation of $S_n$.
\smallskip
By performing the bar-construction on an operad ${\cal P}$, one gets a differential graded operad
${\cal B}({\cal P})$. An algebra over ${\cal B}({\cal P})$ is a (strong) homotopy ${\cal
P}$-algebra. For instance a homotopy associative algebra (often called $A_{\infty}$-algebra in the
literature) is in fact a ${\cal B}( As)$-algebra. Similarly a homotopy Lie algebra
(Hinich-Schechtman) is a  ${\cal B}(Lie)$-algebra.

 Ginzburg and Kapranov have set up a nice Koszul duality theory for algebraic operads, for which
$As$ is dual to itself, $Pois$ is dual to itself, and $Com$ is dual to $Lie$. This
duality is particularly helpful in understanding the cohomology theory of the associated
algebras.

Observe that there exist operads in any tensor category (for instance topological operads).
\smallskip
\noindent {\bf Ariane's thread:}

\noindent  ``Operads: Proceedings of Renaissance Conferences". Contemporary
Mathematics, 202. American Mathematical Society, Providence, RI, 1997. x+443 pp.

\noindent J.-L. Loday,  {\it La renaissance des op\'erades}. S\' eminaire Bourbaki, Vol.
1994/95. Ast\'erisque No. 237 (1996), Exp. No. 792, 3, 47--74.

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