The type of parities is \(\mathbb {Z}/2\mathbb {Z}\), implemented as ZMod 2.

A grading on a type \(\beta \) consists of an additive commutative group structure on \(\beta \), together with homomorphisms

such that \(\sigma (\phi (n))\) is the parity class of \(n\) for every integer \(n\).

For a grading index \(\beta \), is the image of \(n \in \mathbb {Z}\) under the distinguished map \(\phi : \mathbb {Z} \to \beta \).

For a commutative ring \(R\), a graded \(R\)-module is a \(\beta \)-indexed family of \(R\)-modules, implemented as a graded object in ModuleCat R.

An \(R\)-linear graded quiver on an object type \(\mathrm{Obj}\) assigns to each pair of objects \((X,Y)\) a graded \(R\)-module of morphisms from \(X\) to \(Y\).