nLab associative operad

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 $Assoc$ or $Ass$, 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.

Related concepts

• associative operad

• commutative operad

• Lie operad

References

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

• Jacob Lurie, Higher Algebra