Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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.

## Definition

### As a $Vect$-operad

The associative operad, denoted $Ass$ or $Assoc$, 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.