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

where the final summand is the image of the integer \(2-n\) under the shift homomorphism \(\mathbb {Z} \to \beta \).

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

in the grading type.

For a string of objects \(X_0,\dots ,X_n\), an index \(i {\lt} n\) and a degree \(b\), this is the hom-module \(\operatorname {Hom}(X_i, X_{i+1})_b\) of the \(i\)-th morphism in that composable string, in degree \(b\).

An \(A_\infty \)-composition on a graded quiver \(\operatorname {Hom}\) assigns to every positive arity \(n\) and every string of objects \(X_0,\dots ,X_n\) a graded multilinear map

of degree \(\operatorname {shift}(2-n)\). Thus for input degrees \((d_0,\dots ,d_{n-1})\) its component lands in any degree equal to the operation target degree.

Fix an \(A_\infty \)-composition \(m\). For \(r + s + t = n\) with \(s \geq 1\), is the term obtained by first applying the inner \(s\)-ary operation in positions \(r,\dots ,r+s-1\) and then applying the outer \((r+1+t)\)-ary operation to the collapsed string. It lies in \(\operatorname {Hom}(X_0,X_n)\) in any degree equal to the Stasheff target degree.

The sign \(\pm 1\) attached to \((r,s,t)\) is

the product of the Koszul signs of the reduced degrees of the \(t\) trailing inputs. Since the sign character is a homomorphism, this equals \((-1)^{|a_{r+s}| + \cdots + |a_{n-1}| - \operatorname {shift}(t)}\).

The arity-\(n\) Stasheff sum is the signed sum of all valid composites of an inner operation followed by an outer operation, that is, of the Stasheff terms indexed by \(0 \leq r \leq n\) and \(1 \leq s \leq n - r\), with \(t = n - r - s\). It lies in \(\operatorname {Hom}(X_0,X_n)\) in any degree equal to the Stasheff target degree.

An \(A_\infty \)-composition satisfies the Stasheff identities if every indexed Stasheff sum vanishes, for every positive arity, string of objects, choice of input degrees and composable inputs.