nLab
associative operad
Contents
Idea
The associative operad $Assoc$ is the operad whose algebras are monoids ; i.e. objects equipped with an associative and unital binary operation.

One might also consider the non-unital version, whose algebras are objects equipped with a binary operation but not with a unit .

Definition
As a $Vect$ -operad
The associative operad , denoted $Ass$ or $Assoc$ , is often taken to be the Vect -operad whose algebras are precisely associative unital algebras .

As a $Set$ -operad
As a Set -enriched planar operad , $Assoc$ is the operad that has precisely one single $n$ -ary operation for each $n$ . Accordingly, $Assoc$ in this sense is the terminal object in the category of planar operads .

(Here the unique 0-ary operation is the unit . Hence the non-unital version of $Assoc$ has a single operation in each positive arity and none in arity 0.)

As a Set -enriched symmetric operad $Assoc$ has (the set underlying) the symmetric group $\Sigma_n$ in each degree, with the action being the action of $\Sigma_n$ on itself by multiplication from one side.

Similarly, as a planar dendroidal set , $Assoc$ is the presheaf that assigns the singleton to every planar tree (hence also the terminal object in the category of dendroidal sets).

But, by the above, as an symmetric dendroidal set , $Assoc$ is not the terminal object.

Properties
Resolution
The relative Boardman-Vogt resolution $W([0,1],I_* \to Assoc)$ of $Assoc$ in Top is Jim Stasheff ’s version of the A-∞ operad whose algebras are A-∞ algebras .

Relation to planar operads
A planar operad may be identified with a symmetric operad that is equiped with a map to the associative operad. See at planar operad for details.

References
In the context of higher algebra of (infinity,1)-operads , the associative operad is discussed in section 4.1.1 of

Last revised on April 3, 2014 at 05:37:26.
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