Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The associative operad 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 -operad
The associative operad, denoted or , is often taken to be the Vect-operad whose algebras are precisely associative unital algebras.
As a -operad
As a Set-enriched planar operad, is the operad that has precisely one single -ary operation for each . Accordingly, 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 has a single operation in each positive arity and none in arity 0.)
As a Set-enriched symmetric operad has (the set underlying) the symmetric group in each degree, with the action being the action of on itself by multiplication from one side.
Similarly, as a planar dendroidal set, 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, is not the terminal object.
The relative Boardman-Vogt resolution of 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