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.

Terminology

There are many terms used for the associative operad. These include:

• As, in Aguiar & Livernet 2005 for the symmetric associative operad and in Loday & Vallette 2012, Chenavier, Cordero & Giraudo 2018 for the non-symmetric associative operad.

• Ass, in Loday & Vallette 2012 for the symmetric associative operad.

• Assoc, in Lurie 2017.

Definition

As a $Vect$-operad

The associative operad $Assoc$ 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.

Related concepts

• associative operad

• commutative operad

• Lie operad

References

• Marcelo Aguiar, Muriel Livernet The associative operad and the weak order on the symmetric groups [arXiv:math/0511698]

• Jean-Louis Loday, Bruno Vallette, Algebraic Operads, Grundlehren der mathematischen Wissenschaften, Springer-Verlag Berlin Heidelberg 2012 [doi:10.1007/978-3-642-30362-3, version 0.99 pdf]

• Cyrille Chenavier, Christophe Cordero and Samuele Giraudo, Generalizations of the associative operad and convergent rewrite systems, Higher-Dimensional Rewriting and Algebra, EasyChair Preprint no. 143, 2018 [doi:10.29007/mfnh, arXiv:1808.06181]

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