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

together with an \(A_\infty \)-composition, that is, operations

of degree \(\operatorname {shift}(2-n)\) for every positive arity \(n{\gt}0\). No relations are imposed on these composition maps.

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 \(i\), Chain.link 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.

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