nLab associative operad




The associative operad AssocAssoc 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.


As a VectVect-operad

The associative operad, denoted AssocAssoc or AssAss, is often taken to be the Vect-operad whose algebras are precisely associative unital algebras.

As a SetSet-operad

As a Set-enriched planar operad, AssocAssoc is the operad that has precisely one single nn-ary operation for each nn. Accordingly, AssocAssoc 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 AssocAssoc has a single operation in each positive arity and none in arity 0.)

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

Similarly, as a planar dendroidal set, AssocAssoc 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, AssocAssoc is not the terminal object.



The relative Boardman-Vogt resolution W([0,1],I *Assoc)W([0,1],I_* \to Assoc) of AssocAssoc 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.


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

Last revised on November 15, 2023 at 08:27:48. See the history of this page for a list of all contributions to it.