An \(A_\infty \)-precategory over \(R\) with object type \(\mathrm{Obj}\) consists of a graded \(R\)-linear quiver together with operations

of the prescribed multilinear type for every positive arity \(n\).

A chain in an \(A_\infty \)-precategory consists of a positive length \(n\), a string of \(n+1\) objects, and a degree assigned to each of the \(n\) composable morphism slots.

The source of a chain is its initial object.

The target of a chain is its final object.

For a chain and an index \(i\), is the module of the \(i\)-th composable morphism in that chain.

For a chain, this is the graded hom-module that receives the corresponding multilinear operation.

The Stasheff sum of an \(A_\infty \)-precategory is the indexed Stasheff sum formed from its structure maps.

An \(A_\infty \)-precategory satisfies the Stasheff identities when its structure maps define an object-indexed system satisfying .

An \(A_\infty \)-category is an \(A_\infty \)-precategory together with a proof that its operations satisfy the Stasheff identities.