representation, ∞-representation?
symmetric monoidal (∞,1)-category of spectra
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.
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-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.
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.
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.
associative operad
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. See the history of this page for a list of all contributions to it.