Why are the algebras of the associative operad unital?

Question

According to the n-lab page:

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

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

It then goes on to say that:

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

Huh? Where did unitality come from? Sure there's this $1$ floating about, but I thought that just got interpreted as the identity function, as opposed to a distinguished element.

Answer 1

I think something is a bit awry. If you "go back in time" (see previous revisions of the page), you see that first the operad was defined as one whose algebras were associative unital algebras (aka monoids), then someone added that it was generated by a single operation $\Theta$ with the associativity relation.

But if you take the construction of an operad defined by generators and relations (like here), you'll see that if there are no generators of arity $0$, then the resulting operad won't have any operations of arity $0$ (because if $k_i \geq 1$ for all $i$, then $k_1 + \dots + k_r \geq 1$...). So there are two things here:

• The operad $\mathrm{Assoc}$ (or $\mathrm{Assoc}_+$) of unitary associative algebras, generated by $\Theta$ in arity $2$ and $*$ in arity $0$ subject to $\Theta \circ (1, \Theta) = \Theta \circ (\Theta, 1)$ and $\Theta (*, 1) = 1 = \Theta (1, *)$. This is the one that Ittay Weiss describes.
• The operad $\mathrm{Assoc}_0$ of merely associative algebras generated by just $\Theta$ in degree $2$, subject to the previous relation.

These two are not equal, of course -- the only difference is in arity $0$.

You are indeed right to say that $1$ is taken to be an identity operation: it's an operation with arity $1$, and in the endomorphism operad it's the identity.

Answer 2

The associative operad as a non-symmetric operad has precisely one $n$-ary operation for each $n\ge 0$. In particular, it has a $0$-ary operation which gives you the unit in the monoid. The symmetric version of the associative operation (the symmetrization of the non-planar version) has $\Sigma_n$ (the set of permutations on $n$ letters) as the set of $n$-ary operations. In particular, the $0$-ary operations is a singleton set, giving rise to the unit of the monoid.