Let \(A\) be a graded \(R\)-module. An \(A_\infty \)-algebra structure on \(A\) consists of multilinear operations

for every positive arity \(n\) and every choice of input degrees.

Any \(A_\infty \)-algebra structure determines a one-object \(A_\infty \)-precategory, with object type and constant hom-module equal to the underlying graded module.

The Stasheff sum of an \(A_\infty \)-algebra is the Stasheff sum of its associated one-object \(A_\infty \)-precategory.

An \(A_\infty \)-algebra structure satisfies the Stasheff identities when every one of its Stasheff sums is zero.

An \(A_\infty \)-algebra is an \(A_\infty \)-algebra structure together with a proof of the Stasheff identities.

An \(A_\infty \)-algebra canonically determines a one-object \(A_\infty \)-precategory.