2 Gradings
2.1 Grading types
The morphisms of an \(A_\infty \)-category are graded by an abelian group \(\beta \) of degrees. The \(A_\infty \)-axioms use exactly two further pieces of structure on \(\beta \): integer degree shifts, so that the \(n\)-ary operation \(m_n\) can have degree \(2-n\), and a Koszul sign attached to every degree. No degree is ever multiplied, so \(\beta \) need not be a ring. Besides the standard gradings \(\mathbb {Z}\) and \(\mathbb {Z}/2\), this allows bigradings such as \(\mathbb {Z}\times \mathbb {Z}\) or \(\mathbb {Z}\times \mathbb {Z}/2\), in which \(m_n\) has degree \((2-n,0)\).
A grading type is an abelian group \(\beta \) together with group homomorphisms
such that \(\operatorname {sign}(\operatorname {shift}(1)) = -1\); that is, the differential, which has degree \(\operatorname {shift}(1)\), is odd. For a degree \(d \in \beta \) we write \((-1)^d := \operatorname {sign}(d)\) for its Koszul sign.
For every integer \(n\) one has \((-1)^{\operatorname {shift}(n)} = (-1)^n\).
Since \(\operatorname {shift}(n) = n \cdot \operatorname {shift}(1)\) and \(\operatorname {sign}\) is a homomorphism, \((-1)^{\operatorname {shift}(n)} = \operatorname {sign}(\operatorname {shift}(1))^n = (-1)^n\).
The integers \(\mathbb {Z}\) form a grading type with \(\operatorname {shift}\) the identity and \(\operatorname {sign}(n) = (-1)^n\). The parities \(\mathbb {Z}/2\) form a grading type with \(\operatorname {shift}\) the reduction modulo \(2\) and \(\operatorname {sign}(p) = (-1)^p\).
Let \(\beta \) be a grading type and \(\gamma \) an abelian group. The product \(\beta \times \gamma \) is a grading type with \(\operatorname {shift}(n) = (\operatorname {shift}(n), 0)\) in two ways: either the sign is read off the first factor alone, \(\operatorname {sign}(b,c) = (-1)^b\), or, when \(\gamma \) is a grading type as well, the sign is the total sign \(\operatorname {sign}(b,c) = (-1)^b\, (-1)^c\). Neither choice is canonical, so both are constructions to be activated locally rather than instances.
2.2 Graded linear maps and quivers
The graded-linear-map and graded-quiver definitions below do not need a grading type; the type of degrees is arbitrary there, carrying at most an addition. The structure of a grading type is only imposed by the degree arithmetic and Koszul signs of the \(A_\infty \)-axioms.
Let \(R\) be a semiring and let \(M, N\) be \(\beta \)-indexed families of \(R\)-modules. A graded \(R\)-linear map \(f : M \to N\) of degree \(d \in \beta \) consists of an \(R\)-linear map \(f_{i,j} : M(i) \to N(j)\) for every pair of degrees with \(i + d = j\). Graded linear maps of degree \(d\) form an \(R\)-module under componentwise operations.
The composite of graded linear maps of degrees \(d\) and \(e\) is a graded linear map of degree \(d + e\), whose component from degree \(i\) factors through degree \(i + d\). The identity is a graded linear map of degree \(0\). Composition is associative and unital.
Componentwise, this is associativity and unitality of composition of linear maps, once every component of a composite is seen to factor through any middle degree.
Let \(R\) be a semiring, let \((M_j)_{j \in \iota }\) be a finite family of \(\beta \)-indexed families of \(R\)-modules, and let \(N\) be a \(\beta \)-indexed family of \(R\)-modules. A graded \(R\)-multilinear map of degree \(d \in \beta \) consists, for every choice of input degrees \((d_j)_{j \in \iota }\) and every output degree \(i\) with \(\sum _j d_j + d = i\), of an \(R\)-multilinear map
Graded multilinear maps of degree \(d\) form an \(R\)-module under componentwise operations.
For a commutative ring \(R\), a graded \(R\)-linear quiver on an object type \(\mathrm{Obj}\) assigns to each pair of objects \((X,Y)\) and each degree \(i \in \beta \) an \(R\)-module \(\operatorname {Hom}(X,Y)_i\) of morphisms of degree \(i\) from \(X\) to \(Y\).