nLab
associative operad

Idea

The associative operad Assoc is an operad which is generated by a binary operation $\Theta$ satisfying

$$\Theta \circ (\Theta ,1)=\Theta \circ (1,\Theta )$$\Theta\circ (\Theta,1)=\Theta\circ(1,\Theta)

This property of the operation $\Theta$ to be associative is not to be confused with the axiom of associativity imposed on every operad.

$\mathrm{Assoc}$ is hence the operad whose algebras are monoids; i.e. objects equipped with an associative and unital binary operation.

Definition

As a $\mathrm{Vect}$-operad

The associative operad, denoted $\mathrm{Ass}$ or $\mathrm{Assoc}$, is often taken to be the Vect-operad whose algebras are precisely associative unital algebras.

As a $\mathrm{Set}$-operad

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

As a Set-enriched symmetric operad $\mathrm{Assoc}$ 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, $\mathrm{Assoc}$ 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, $\mathrm{Assoc}$ is not the terminal object.

Properties

Resolution

The relative Boardman-Vogt resolution $W([0,1],I_{*}\to \mathrm{Assoc})$ of $\mathrm{Assoc}$ 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