Given input degrees \((d_0,\dots ,d_{n-1})\), the output degree of the \(n\)-ary operation is

where the final summand is interpreted using the grading shift .

The common degree of the arity-\(n\) Stasheff relation is

in the grading index.

For a string of objects and prescribed degrees, is the \(i\)-th graded hom-module in that composable string.

For a composable string of objects and degrees, this is the graded hom-module in which the corresponding multilinear operation takes values.

Fix object-indexed operations \(m_n\). For valid indices \((r,s)\), is the term obtained by first applying the inner \(s\)-ary operation in positions \(r,\dots ,r+s-1\) and then applying the outer operation to the collapsed string.

For a valid pair \((r,s)\), this is the parity

computed in .

The integer sign attached to \((r,s)\) is \((-1)\) raised to the parity recorded by .

The arity-\(n\) Stasheff sum is the signed sum of all valid composites of an inner operation followed by an outer operation.

An object-indexed family of multilinear operations satisfies the Stasheff identities if every indexed Stasheff sum vanishes.